Puppy nipping/teething is completely normal and to be expected. It is not to be tolerated. Any puppy allowed to nip and chew on it’s owner will develop a habit that later will be a nightmare to break.
Initially, substitution should be used. Just like a young child, offer the puppy something better than you to chew on. If the puppy is determined to make you the “new chew toy”, put the puppy down with a firm “NO!”. Keep a squirt bottle handy so if the puppy will not “go away” when placed on the floor, you can squirt puppy a couple of times to make your point clear. If you consistently put the puppy down every time it starts chewing on you with a correction word, puppy will correlate between the attention that it receives and the rejection it receives.
Because they love attention and being held when they’re little it is probably the strongest message you can send. I often hold a real beef bone (that has been parboiled of course) and allow them to chew on that. Any time that they move their mouth to work on my fingers I move the toy into the way so they have to chew on the bone and not my hands.
I do not allow puppies to chew on me at all!