My chin used to do this too! But I put an end to it by picking her up by her tail whenever she nipped my arm. I would only do it for a couple seconds (I know breeders handle their chins this way but to me it looks like it would hurt) and I would say to her in a firm-but-non-yelling tone, "Chloe, that's a no-no. I know you want down but you hurt my arm when you bite me."
Then I would go back to holding her and if she nibbled me again I would hold her by her tail and give my speech again. Eventually she got the idea.
I try not to hold her for too long because I know how much she enjoys running around my room, jumping off her boxes!
Also, maybe your chin didn't like the position you held her. It took me a while but I realized that Chloe is most comfortable on my shoulder, peering over my back with my hands supporting her back legs. She really loves being petted once you get her in the right position!
Also, I give her a peice of a raisin every time I take her out and hold her so she associates being held and petted as good.
Now she practically hops into my lap to come out because she wants her treat! It takes time and patience...
I'm not a chinchilla expert or anything, but hopefully these ideas help. Maybe someone with more experience has some suggestions?