Yes. I've had experience working in two small communities that had postal service, but no banks. One is Cobalt, Ontario, and the other is the Inuit community of Ulukhaktok in the Northwest Territories. If there were banking available through the post office in these two places, then it would greatly ease the burden on people to get on with the day-to-day business of life. They look forward to the mail arriving, but they really have a great deal of difficulty getting their banking done, and these places don't have good broadband access.
In Cobalt, Ontario, the population is aging, and even if they had broadband access, they wouldn't feel comfortable with it. They would prefer to deal face to face with a clerk.