It will take him a while to learn you mean business when you stop playing. He's trying to get your attention back by stepping up his play...in this case, stepping up the biting. I noticed the same thing when raising puppies so I took to stopping play and distancing myself from the little tyke. Normally we'd be playing in a puppy-proofed room that was gated off, so I'd just step over the gate till puppy had calmed down some. He'll get to the point where he connects the biting with cessation of play but it'll take a little time.
The worst of it for me is always the hamburger hands--they get pretty raw after a while till puppy learns that chewing on them isn't a viable option. To get through the worst of that phase I started using bicycle gloves--the kind that leaves your fingers uncovered but covers the back and palm of your hand. I find my patience with puppy shenanigans stays much stronger when my hands aren't at risk of puppy gnawing.
No denying that raising a puppy is not for the faint-hearted
, but they're so worth it!