Learn English – Another word for “designated”

If you would not consider rural areas to have negative connotations (and faraway places are unlikely to be urban cosmopolitan), then the Australian words wayback and outback are applicable.

Another word is outlands

the outlying districts or remote regions of a country, provinces: a foreign land: a name unknown in the outlands.

Endless mode is commonly used in video games, and so is survival mode, which sounds a bit more exciting – how long can you last? Infinite is also sometimes used.

Of course, you can call it what you like. There's no single correct answer. You could even make up a new name like panic mode if it fit the atmosphere of your game. A more restrained suggestion like untimed might work best if it's a relaxed game with no pressure – play as long as you like, it's untimed!

So really, it's up to you as a game designer to pick the term that fits your game best. That said, my personal favorites are survival and endless. I usually end up using one of those two terms when I'm talking to my friends, no matter what the author of the game chose to call it.