sharing my recent experience in banking with a PR and no South African ID, only tried with 3 banks fwiw:
- Tyme is a no-go, they only have online / app onboarding which starts with an ID number
- Capitec got my blood boiling, staff at two different branches didn't seem to have a clue what a PRP even is and insisted on seeing a work permit ... once you get past that obstacle they insist on a SA ID, I've confirmed with their client care center which had a more informative reply, but still not helpful in our case, you must have a valid port and residence permit, but also a salary slip / employment letter i.e. it can be done without an ID but then you must be employed, or it can be done without employment, but then you must have an ID
- FNB sorted us out immediately, no problem, without any nonsense about work permits or employment or needing an SA ID