If the servers are down or your PS4 is not connected to the Internet, no other parts of the game are available and no progress can be saved. (This means you'll lose any in-game credits or level progression earned in Arcade mode.) Obviously, online matches depend on the servers. But so, too, does the rest of the game—stuff that has always been available offline in the past. Campaign mode (the career mode that includes the notorious Gran Turismo driving school) is off limits while offline. Also unavailable offline: buying new cars, viewing your garage, editing car liveries, and even the "taking photos of fancy cars in exotic scenery" mode.