Mine's a 6SPlus. It seems to be a problem that depends on what mix of apps you've got as well as other unknown factors and it's apparently a problem that goes back over a number of IOS updates. From what I've read, it's something to do with the OS re-indexing and working flat out in the background.
Anyway, the answer was backup, factory reset, restore from backup and it seems fine now.
Shame that not one of the four support agents I spent time with yesterday didn't just say, "Yes we are aware that this happens in some cases. What you need to do is this, this and this."