Get small means become small. For example:
- Is it just me, or are fine prints getting smaller and smaller?
(fine print = information containing details written in small font size) - Look how much your daughter has grown!
- Yes, she’s not getting smaller, that’s for sure.
As getting small is about physical change, a part of me got small can only be meant figuratively. It is not a fixed expression, so you will need to create some context in which it makes sense. For example:
Maybe this person achieved something which he was really proud of, but then he realized it wasn’t his abilities that helped him succeed, he was only lucky. So he felt disappointed in himself and felt that his achievements have been diminished. At this point, he may have said “a part of me got small” to illustrate how he feels less proud than before.
