Yes, Fortnite servers are currently down, try not to panic

Down and out.

PlayStation Store

According to Epic Games, Fortnite is currently undergoing a period of “degraded quality” that will be affecting all players. You may have difficulty logging into the game or getting into matches because of this. You might also have issues with your Locker or receive messages telling you that you do not have permission to play Fortnite. At the moment, it is thought that all these different issues are caused by the same problem.

According to the Fortnite Status page, Epic is aware of the problem, and is working to fix it:

Update – We are still actively working to resolve these issues.

Please note that if your Friends aren’t listed in your Friends List, they will be there again when these issues are resolved.
Mar 17, 19:23 UTCInvestigating – We’re currently investigating issues with logins, matchmaking, the Item Shop, parties/friends/messaging, and game services. We’ll provide an update when these are resolved.
Mar 17, 18:46 UTC

The issue is also affecting features like your Friends List, so if you log on and don’t see anyone on your list, it will be resolved as soon as the overall issue is fixed. There really isn’t much you can do about it from your end, as the issue is occurring on the game servers, it is up to Epic to resolve it as soon as possible.

It is not known if the issue is somehow related to the 12.20 update which arrived earlier today and introduced helicopters, and the new Limited Time Mode called Spy Games.