Yes, we do. You're correct that in British English have got is what's used, but at least in American English, we do say have gotten.
I've gotten 10 parking tickets already this year!
I have gotten almost no sleep since my baby was born.
In American English we use have gotten both when writing and when speaking. As Peter Shor notes, it sounds very odd to us to hear Brits say (or write) have got!
You might also be interested in this question on EL&U which asks about the difference between the two constructions and has some interesting answers.