Puppies have a LOT of energy and one of the ways that this energy is 'spent' by them is through oral play. If pups are left in their litter the recommended 12 weeks, much of this nippy play is reduced because the mother and the other pups teach each other that harsh mouth play is not acceptable. Since you are now  the playmate, you must teach it that oral play is not acceptable.

    The fact is that you are not going to be able to 'stop' the oral play, (it's necessary to the pup's development), so you must RE-FOCUS it to 'legal' toys (NOT your hands or any other body parts). This means having something 'legally chewable' nearby at all times for starters. Stuffed toys are great (buy them for 10-25 cents at yard sales and when they get dirty or badly torn, just throw them away w/o worry :)

    When pup starts attacking/biting in play, replace your hand in his mouth with some legal TOY or chewie  for him to chew on. Admittedly, pups prefer hands, toes, etc. but this is, as you've already found out, not good. Some use large rawhide chewie bones to help pup in this teething/play process. Some fear they are too dangerous ... that a pup can  get a piece caught in the throat and choke.  Do note that the eyes, nose, whatever is plastic on a stuffed toy needs to be removed as the pup can definitely choke to death on these dangerous extras. (Just sew up the  eye socket place after removal of eyeball so the stuffing doesn't pop out).

    Some people who get a wee Bichon pup don't realize that these pups may look like toys, but they are indeed Dogs ... and accordingly need the same level of activity that all dogs require and that is a LOT of exercise. He won't freeze outside in the winter if you're moving and exercising ... in fact, he'll LOVE it :))
And though you need to be very cautious about getting your puppy too hot in the summer, early morning activity or late evening walkies will quickly become his favorite time of day.

    I'd also recommend getting your puppy and you into Puppy Kindergarten asap and continue with obedience classes.

    Key: Re-Focus that great energy toward legal gnawing and that problematic biting will soon be a problem of the past.

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Temperament