I haven’t been able to find any data on this, but concensus among faculty I’ve talked to is that it is possible but extraordinarily rare. There are probably a few reasons for this, among them:
- Getting a Ph.D. takes a long time (4-7 years) when you are working at it full time. Doing it part time would take, presumably, much longer.
- Students usually get paid to do their Ph.D., so working is not as necessary.
Finally, as an advisor, I’d be hesitant to advise a Ph.D. student part-time. The project they would be working is my project too, and I want to make progress as quickly as possible.