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DavidT8899 Leisure World is a private retirement community where people live full-time. Recently, there was a push to incorporate it as a separate city, but I don't think that succeeded. Same with a gated community here in Savannah, "The Landings," which has discussed incorporating. If it did so, it would be restricted to people who lived there and their invited guests.
I'm sure there are some cities like that, but I can't find any. Ryderwood in Washington began as a logging camp, was converted into a retirement community, then eventually took over the entire city, but I don't know if access is restricted in any way. What you're looking for is a restricted-access development that incorporates into a city.
"Sundown towns" (cities where Black people could not remain at night) arguably would have been restricted-access, but wouldn't have been privately owned.