Fur biting is not as straight forward as seems, everyone generally has a good idea of 'how' it happens but never seem to suggest any useful information on how to 'stop' this habit.
First of all, if you suspect your chinchilla is fur biting, then it's best to go with the check list shown here below in the grey box because there are different reasons a chinchilla may fur bite
The check list will help you identify if your chinchilla has a problem that can be cured to a reasonable extent - or if more serious - a vital check out at your vets would be necessary
These are the 4 main issues on 'fur biting' I will discuss in this article

Fungus infections (as known in Trico-phyton Mentagrophytes) is quite different to fur biting, to the trained eye you can easily tell the difference between fungi infections and fur biting, with fungus the first areas to be affected are round the eyes and nose, sometimes the tail, whiskers also do break off around the nose area, these are early signs that your chinchilla has a fungus - the more advanced version is when their fur falls out and the difference is quite obvious because the chinchilla will have bald patches where fur should be and the skin scales and flakes away - with fur biting it does not 'fall out' it looks 'chewed' in a uneven like pattern - the picture of the chinchilla at the top of this article is a good example of how bad fur biting can get. With fungi infections outbreaks can occur in the Autumn and Winter because of the damp conditions that cause the fungus to grow and spread, your chinchilla must definitely be kept away from any possible damp areas - Fungus and other infections that chinchillas can obtain will be dealt with in more detail in another article on this site later on in the future
 Hay is another common cause of fungi, hay left in the cage too long and in the wrong conditions is a sure sign the chinchilla will get an infection - hay is a carrier of fungi as well as been their food and this counts for the pellet too - water must be changed daily and bottles cleaned regularly.
Early signs are easy to deal with, you certainly must go to your vet first to confirm it is a fungi, if it is, then the vets generally give you a fungicide (like athletes foot powder) which can be mixed in the chinchillas dust bath, Laboratory tests are also advisable as it is important that ANY skin fungi - wether it be on chinnie or you - can be infectious!
PREVENTION - clean hygienic cages and a tablespoon of athletes foot in the dust bath - changed every two month
Not to be continuous - just add the powder occasionally / about 2 months after the fungus has gone in case of risk of it lingering after

This is a serious problem with all rodents not excluding chinchillas, I will write about Malocclusion on another sheet  - but first - what has malocclusion got to do with fur biting?
Just because a chinchilla fur bites does not mean that it is a sure sign of toothache, in general fur biting is a hormone/hereditary problem (which is mentioned below this paragraph) but when a chinchilla does have Malocclusion, especially advanced, then pain can cause the chinchilla to be distressed, which in some cases makes them fur bite, not all chinchillas that have teeth problems fur bite - just a few
Always check your chinchillas teeth, malocclusion can come at any age, when a chinchilla fur bites over this problem there are more obvious signs to look for, mostly know as the 'slobbers' and from that expression you can easily see what I am getting at - the chinchilla will dribble down the front, eat awkwardly (probably not eating much at all) if you fear this is the case you definitely must take your pet to the veterinary surgeon for a professional check up and probably an X-ray
I will be writing a sheet on how to identify and check for malocclusion in the near future

Many chinchilla owners with experience do dispute the fact that most of the reasons that chinchillas do bite their fur is through a hormone disorder, well, from what I have read, this can only be so with certain cases - for instance:
If the chinchilla is female, it's more than likely a hormone disorder especially if pregnant, not all pregnant chinchillas will fur bite, and if they do, then generally it stops once the young kits are born
Sometimes afterwards the mum - especially first baby mums - may have a little too much maternal zeal and will nibble the fur on the kits head or back, but obviously this stops after the kits grow and leaves
Sometimes young kits have this habit when separated from mum after weaning or as they reach maturity, it doesn't happen often but if it does it's generally a temporary situation
As for a dietary deficiency, it's most unlikely that if your chinchilla is eating good basic pellets and plenty of quality hay along with water and not too many treats, then dietary does not come into it.
Irritations to the chinchillas skin can also cause temporary fur biting until the problem is solved, they may even fur bite along with the fungus mentioned above - giving the impression that it's an advanced infection! but it generally is irritation of sand baths, the wrong dusting powder, sometimes too many baths - certain chinchillas don't need too many baths - it really depends on the weather if it is quite warm (especially in warm climate countries) and how much they are handled

If you have read the paragraphs above and your chinchilla does not relate to any of them, then it's more than likely this is the part you will need to read
Firstly, now that you have identified your chinchilla to be a 'true fur biter' do not despair - just because furball bites his fur doesn't mean he is not well, if it's none of the above mentioned - it's definitely in the genes!
And if they are PRONE to biting their fur then the only problem will be cosmetic, no health problems will occur from biting fur - just cosmetic problems - true it looks unsightly and some can be worse than others at biting but I tend to look at it from a human version  - NAIL BITING - once a nail biter always a nail biter!
FUR BALLS 
Some people panic and compare it to cats and dogs, and that creating 'FUR BALLS' could be fatal, but from what I have read on chinchillas and fur balls there definitely is no reason to panic, one certain modern chinchilla book does seem to indicate that 2 autopsies made on their chinchillas found fur balls and came to the conclusion that this could be the cause, I know of those that have bred many, many chinchillas and tend to disagree - 2 fur ball victims out of thousands of chinchillas does not prove that fur balls can kill, but having said this, there is always someone to disagree with me, they are entitled to think what they belief - but unless it can be absolutely proven (and we are talking many years of breeding chinchillas all over the world here) I will believe that fur balls do no harm to chinchillas
GENES
This is generally the real culprit and cause, it can be a disaster to a breeder who produces 'show quality' animals, he definitely does not want that 'gene' flowing through his herd and if he sells as a pet, or to other breeder new or old members, then reputation is his name, when buying a chinchilla from a GOOD ESTABLISHED breeder or hobby breeder, it's a sure guarantee that your chinchilla will not fur bite - although there is always the exception he might - some bad genes 'jump' a few generations!
When you buy from a pet shop that is an entirely different thing, I would say that of all the pet shops in the UK 99% of shops will buy from small hobbyist with little knowledge or unscrupulous breeders (those that do not care what gene is carried through as no one knows it's them through a shop) not a good guarantee then, when you go shopping, there are a few shops that breed with quality - pet shop quality no doubt - but little chance of them breeding fur biter s- rare shops - but out there! and also you may be one of the many lucky owners that have a fur biting free chinchillas from a shop - it's just a form of chinchilla 'Russian roulette' really!
CURE
First of all, before you get too excited over the title here, let me flatten your hopes a little and say that until now - there is no 'cure' - but to give you hope, there is a way of 'controlling' this problem
Just Like I mentioned above - you can class it has a human form of nail-biting
Some chinchillas will hack away until they are completely bald and others will be what you call 'spontaneous' biters,  in other words, they will chew their fur large or small, then leave it to grow, then a month later or so decide to chew again - spontaneous chinchillas are generally the best to control
Now this is coming from personal experience, the first two chinchillas I bought were pet shop buys, like most people who buy their first chinchilla, it is usually a shop buy until you find out about breeders!
These two were perfectly OK until they were two and a half years old, two females, one beige and one grey - fur biting can start at any age, any sex and at any time! 
Garbo my grey chinchilla started first - we had a downstairs bathroom decorated and as their room (the granny flat section) was near to this bathroom, along with the banging and radio noises occurring, Garbo had just had enough and began to chew her flanks (generally the starting point on all culprits) Toffi-nut my beige did not chew during this period
Considering the fact that 'noise' was the key starter of her fur-biting I did not panic - keep that in mind, some chinchillas do not mind noise, others are more sensitive and may chew in response
Here is a photo of Garbo looking rather ashamed of herself

Note the small starting point on her flank? well, that can get bigger on both sides, then she stops
 spontaneous fur biting is what you call it!

When the workmen finished, we had a lovely new bathroom but Garbo did not look so lovely
AND she did not stop there, this had been in her genes since birth, and Toffi-nut her sister? would she have it in her genes too? of course! but in an entirely different method
Once there was peace and quiet in the house I reckoned I would not see this happen again, I also had another pet shop chinchilla - a Wilson white and three more breeder chinchillas at that time - not one of those 'nibbled a plunket' all through the noise made in the 2 weeks of work!
Garbo slowly chewed through until she looked pretty sad, living with Toffi-nut her sister didn't help, fur chewing can be contagious, and Toffi-nut gleefully followed her sisters 'fur steps' and started to chew herself, not only that, she chewed Garbo! I knew it was Toffi-nut because Garbo's chewed fur appeared on parts of her body that she would not be able to reach herself - including her whiskers - guilty Toffi-nut
And what about the chinchillas personality? - does that make a difference to one that fur bites and one that doesn't? - not really - although in very good breeder dispositions - yes
Going back to these two girls, Garbo has a sweet nature but it did take me a while for her to get used to me, she is the nervous type, but on the other hand, Toffi-nut is the 'wild one' of the two, has a very tomboy style about her, possessive and a bully with food, toys, treats and other chinchillas, a great attention seeker too and very adventurous, but Garbo is her own person and she can 'control' Toffi-nut quite well, she is more stable - like an the older sister
they both still fur bite, that is something I have to 'control' with them often, it takes patience and an understanding of their personality to help control this problem they have
NOT all chinchillas will fur bite all the time - some may just do it once or for a short period of time between a month and a couple of years - and then never again, it generally is caused by upset, once the cause of upset is removed, then chinchillas in some cases generally stops
But for those that do not, I have suggested ways here that have helped me in the past and present

NOISE
As mentioned above - unusual noises, loud or discreet can cause panic in fur biters, changing their living quarters to another part of the room can cause quick nibbles to the fur and even a new environment all together like when you bring them home from either pet shop or other, some may chew with change of scenery but should stop as soon as they settle down (may take a few days) remember, chinchilla fur, once bitten WILL grow back no matter how bad they chew, it just takes a long time, three months is the general length it takes and if they chew for a weeks then you will have to look at your moth eaten furry for another few weeks before you know for certain he has stopped
CAGE
 Small cages can cause fur biting - generally a problem in large breeding quarters where the cage is too small, some chins don't care but others don't like it - it's obvious that the largest cage you can accommodate is the best cage - a large cage with too many chinchillas can be a cause too, especially if one doesn't get along with another and may bully the 'fur biter' - keep a check on your chinchillas - make sure there are no bullies! A cage size of 3ft, 18" depth and 2ft high is perfect for one chinchilla, good for two if they have a little playtime and not too good for three unless they have a massive playtime - I have 3 chinchillas in a 55" high, 40"wide and 22" deep cage - and I plan to add another in this one, then that's it, they get 1 hour playtime or more a day when possible
Having said that, Garbo and Toffi-nut live in a cage of 4ft high, 18"depth and 3ft wide, so obviously something else that causes the fur biting? yes - read further down to find out!
BOREDOM
This is rather obvious, wouldn't you chew your fur if you had nothing else to chew? and shelves can be a bit boring if this is all you get! EVERY chinchilla MUST have chew toys, oak, apple wood, pine (untreated) and most fruit trees are the perfect choice, go into the woods for chinnie and collect some free!
All they need is a good wash in salted water, rinse and dry - it will last them ages, wether they can hack through a cable wire in second or not
Loneliness can cause fur biting - but if you give your pet plenty of personnel attention there should be absolutely no problems here

Now having said that, as I noted out to you before, big cages, friendship with other chinchillas, no noise or minimal noise, plenty of chew toys - and they still fur bite?
This is true with my own 2 little monsters....but one day, I gave them some 'new' toys, some apple branches I had in my field, and what happened, they stopped chewing , and their fur grew for 4 months - but that wasn't the end, after 4 months they started again, bear in mind that spontaneous biting can be a gap of 1-3 months while fur grows, they chew another area or it gets bigger in the same area, you get to know if they have stopped or started
The answer lay in 'occupational therapy'
OCCUPATIONAL THERAPY
Sounds rather odd doesn't it? but I think anyone with half a chinnie brain will know what I am getting at
It's similar to boredom but without fault of the owner, they can have big cages, lots of chew toys, playtime and a no noise zone area - and what do they give us after all that?  DEMAND!
Demand for attention is a bit like babies, no matter what attention you give them, all the toys you throw into the pram and what do they do? scream and wail and DEMAND MORE!!!
Chinchillas do not scream and wail (okay, the odd squeak of protest) but they can demand more, if you have a few chinchillas like me and the chinchillas2home team have, it's very hard to give one set of chinchillas extra attention, and also make sure the others are not left out, this is a lot easier if they are your only chinchillas. 
But demand my two wanted and so to work I did go, first I already had a wheel in one of my other cages, but it would of been cruel to take it away as they loved it so, so I bought some more, in fact I bought 4 for future usage on other chinchillas, you don't have to go that mad, but a wheel is a great toy for chinchillas, and the best controller of fur biting I have seen yet! but it isn't always the FINAL cure, just one part of it (the well know flying saucer wheel is the best!)
The wheel did occupy the attention seeking fur biters for a while but it's not enough, they need different chew toys periodically, the other chinchillas will chew until there is nothing left, same with these two, but they get bored with the same sticks and pumice stones, variety is the spice of life to them, I give them one set of toys one week (the odd cereal box works too) and another set next week and so on, swap sets too! don't throw them away half chewed, they will still look at it as 'different'
 I find that this is always a very good method, if I forget for around 3 weeks - then lo and behold they will 'chew' their fur again very soon afterwards - but then stop as soon as you give them this attention
Playtime is another good one, half an hour to one hour a day at the most, playtime MUST have different boxes put in front of their noses, not as often as chew toys in the cage, but the odd occasional cereal box works a treat - even an old book! although mine are limited to one room, sometimes I let them 'escape' for 5 minutes - fully observed by me in case they decide to dig their 'pearlies' into the cable wires, furniture and skirting board of course!
this helps Toffi-nuts adventurous spirit and helps Garbo build up a bit more braveness
TOO MUCH HANDLING can cause chinnie fur biting - Some chinchillas just do not like been picked up as often as others, it may take a while even years before your pet really doesn't mind been picked up and handled, always cuddling, kissing and holding for very long periods - 5 minutes to certain sensitive chinchilla is a lifetime! this is why children must have the attention of an adult nearby - they are not really a children's pet, chinchillas are semi-domesticated - still a little wild animal
You mean you do that and you're an adult? okay, then stop it, if you're chinchillas a fur biter then it might be too much handling! -  I know it's tempting to cuddle your fluff and some chinchillas don't mind it - but really, if your chinchillas is one that does get upset over been held then the trauma they are going through inside that calm exterior 'look' they have - I am sure you would stop straight away, this doesn't mean you have to never hold them ever again - it takes time to build up a relationship - just like boyfriends and girlfriends - the longest friendships are the best friendships and the most trusting ones! and chinnie should be with you for a long time yet - remember -  15 years to us is 80 years to them, and how long does it take for you to build up a 'good' friendship and 'trusting' friendship? 
it takes more than just a year doesn't it?
Well expect that from furball
TOO RICH A DIET can cause a deficiency and create skin irritations, or too much humidity in the climate
Having said all this, if everything on this page fails to stop your chinchilla - and it will take around 1-2 years to really know - then unfortunately you have a rare case of furry selfishness on your hands
As long as your pet has the best diet going - plain pellets (at least 15% protein content) good hay or grass regularly along with fresh water every day that's really what matter, if you have a rather grubby looking but healthy and much loved chinchilla - who cares what he looks like!

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