You are likely pressing too hard on the strings.
Press only as much as you need to make a clear sound;and don't hold onto a chord too long! When we practice, we guitarists sometimes tend to do these things.
And don't ignore pain. It's nature's way of telling you something is wrong. Fingertips have loads of nerve endings. And hands are marvellous instruments in their own right---treat them with respect. You'll avoid accumulating problems with tendons, ligaments and such.