Earlier quoted context omitted.
There have been suggestions over the years for the post office to offer banking services. They have branches literally everywhere, they're independent yet still a government agency, and have no profit motive.
I could not imagine the amount of financial, operational, technological and regulatory infrastructure that would need to be ported over, however.
Above examples are just passbook (personal) savings and misc services. Super simple.
Though I'd be open minded towards some overlap with credit unions.