This is kind of what happens in Las Vegas, but to a lesser extent than the City of London. Most of what we think of as "Las Vegas" is actually the unincorporated territory known as "Paradise."
That's pretty common, though? It's the whole reason the census has the concept of metro statistical areas. Phoenix, Portland, Seattle, all have this same effect to varying degrees.
But none of them have a rule saying the POTUS cannot visit with the Mayor approving, so there's that. And "City of London" is tiny compared to the rest of the London metro.
