Why Does My Kitten Lick My Fingers?

My kitten has been doing this for a while and I don’t know why. She licks my fingers and then bites them like she is trying to get at the taste of something on them. Then when she does it, it’s almost as if her tongue touches my skin in such a way that makes me shiver with pleasure, but also hurt slightly. It doesn’t seem to matter what type of finger or hand I use when she licks me; whether its my right or left hand or even both hands! Also, when we first got her (about 10 weeks ago) she was licking everything whenever we entered our room and would stop only after I’d petted her head so much that it started to hurt! What could be wrong??