Recently suggested on reddit, a Limited Time Mode with a Gun Game theme might be coming to Fortnite Battle Royale in the future based on a response by Dusty Devo, one of the developers at Epic Games:
Interestingly, the idea was already on their list for future content and they seem keen on trying an LTM that has a progressive weapon loadout. The ‘Gun Game’ mode could feature a smaller play area with the ability to respawn, something we haven’t yet seen in any mode available in Battle Royale.
It appears they’re currently undecided on whether you would progress from stronger weapons (Legendary Scar) down to weaker weapons (Common Pistol) or vice-versa.
Either way, we’re very excited to see how this is implemented if it ever makes an appearance, it’ll be completely different from anything we’ve seen so far.