I actually have a horror story about bitten fingers. At one point, I had three male chins, but they all played separately. However, I wanted them to get used to each other so I let each one play around the other two cages. Well, one night Binkie was holding onto the bars and Fatty (the new boy) bit one of Binkie's little fingers. I tried to keep it cleaned but figured it was minor since it didn't seem to bother him. I went out of town for like a day and a half and had a friend check on them. When I came back, Binkie had chewed his hurt finger off and his whole hand was purple and swollen. So we went to the vet who wanted to amputate his hand because she thought the infection might spread to his bone (Now having worked around that particular vet, I knew that on occasion she could be a little quick to amputate) I asked if antibiotics would have a chance at healing it and she said yes. So I gave him IM baytril for a week or two. He seemed alot better. Two days after we went for a check up and stopped the baytril (vet said he was fine), i came home to Binkie gasping (or rather breathing like an animal thats dying). We rushed to the ER and they said he was septic. Now I don't know if that came from his hand or from the fact that he had a permanently prolapsed penis (which I always worried), but still. he didn't make it.
The point of that story is to say just keep an eye on the finger for swelling and redness because unfortunately, it seems like infection can spread very quickly in chins (well probably in general but chins are so small).