It actually sort of sounds like a callous. I used to get it on the tip of my left index finger from writing. Do you write with that hand? Mine only peeled when I would pick at it though. But, I do have callouses on my feet that crack and peel now and then. As for my index finger, mine would stay dented in for moment when pressed. Have you tried moisturizing? You might try using vasaline. I would use a lightly frangranced or fragrance free lotion first then put a dab of vasaline on it to lock in the moisture and, though it sounds funny, either where a sock over that hand or a looser fitting glove. Something that won't rub off the vasaline too quickly and will let your skin breath just enough yet still lock in the moisture. See if that works. Give it some time. Rome wasn't built in a day and skin problems don't just get resolved in a day either.
Take care, I hope this helped, and keep in touch!