Same here. Only even bitten once, but her teeth went right through my finger nail and into the finger.
This one came into the pet store and was super skittish. I normally don't like tail grabbing as none of mine needed that, but with this one it was necessary. We'd had some people on a waiting list for a certain sex. So I popped back into the kennel off the floor to open the habitat from the back. I knocked gently on the door to let it know the back was going to open. Of course, it freaked out and got in a corner and was actually growling at me. I've
never had one do that. So I talked calmly, grabbed, checked, it was a male and turned it loose, and closed the door. The owner was in the back hand-feeding baby parrots and of course, was like, "Why don't you get it out and teach it, it is okay to be held?" My eyes could have rolled all the way up into my head. If I'd been doing that when he walked back there to start feeding he'd have screamed at me for being a manager and not on the floor. So I caught the stupid thing again and kept trying to get in on my arm in the crook where it would feel safe. I never let go of the tail and every time he touched down on my arm he'd try to leap off. The fourth or fifth time I was bringing him back down and he nailed my finger. He missed the bone but went through the meat of the tip, opposite the nail. You could have slipped a barbell (like piercing) through the bite. I was gushing blood everywhere from my left hand, holding a white chinchilla in the other, and staring at the owner who is just staring back at me. I asked him to help and he threw a towel at me.

Somehow I managed to get the latch flipped, the door open, and the chin back in the habitat without getting blood on him. The owner decided no one was allowed to hold the chin until his wife took a look at it and put a sign on the back of the door. She came in the next morning and per the usual, rolled her eyes, made some disparaging remark about my ineptitude, and snatched the door open. The chinchilla launched itself straight at her face. She ducked, he landed on her back and escaped and they spent two hours trying to catch him. He got put up with a sign that said only I was allowed to feed or water him until the supplier picked him back up.
Ouch.
Is there any way you can tell when they're annoyed and wanting to bite? I'm sure they'd bite out of fear, but is there anything else like to cause them to bite?
Chinchillas do little test nibbles just to see if stuff is edible all the time. It doesn't hurt but can shock you. It's why I normally don' advise long acrylic nails for chin owners because they seem to love the texture of the nail, but you may not feel them biting through it until it's already happened. lol If the chinchilla is really trying to get away from you, growling or making any other threatened noises (not the dying rabbit noise they make for fun at night when you're sleeping and they are doing their little chinchilla raves), stomping/fidgeting with their front feet, ears back, whiskers back, etc. It's pretty obvious if they are pissed off.