It's a bit old. But I haven't been around here for a while... Actually mBank is strict. I tried to open an account there and they require: PESEL and Karta Pobytu just to open an account with them.
ING requires Karta Pobytu to give you a credit card. And I think this is the case with most banks (Or financial stuff).
Archives - 2010-2019 / Law / Is there is any Poland's bank which offers credit card to foreigners on work permit and visa