Aside from UAE (not even there so much nowadays), I've only seen banking for non-residents in that region for people or businesses with strong local connections, the occasional HNWI seeking private banking, and specific expat communities. Bahrain is no exception.
Notably, Al Salam Bank is a Bahraini bank which bought a failing bank in the Seychelles, thus creating Al Salam Bank Seychelles, as a way to offer international banking.