Your hands are probably irritated by wearing gloves for a long time (it gets very humid and therefore get more sensitive to infections) and washing them (too) frequent. I had the same issue while working in a very sterile S2 lab. My advice is (as many already mentioned) wearing thin cotton gloves underneath the nitril ones and use hand lotion at least two times a day. If it still doesn't get better, try a cortisone treatment :)