Visa Waiver Ban
The K-1 visa, when issued, allows the fianc of a U.S. citizen to travel to the U.S., and the marriage must take place within 90 days after entry. Then the foreign spouse can apply to adjust status to that of a permanent resident (green card holder).
Having a green card allows for more freedom and options than a P-1 visa, and it's valid indefinitely, as long as the holder adheres to its provisions. (P visas, with renewals, are good for up to five years.)
Be advised, though, that the K-1 visa can take several months to issue. P-1 visas can be applied for using the premium processing service available from the U.S. Citizenship and Immigration Service (USCIS), for an additional fee of $1,000. This guarantees service within 15 calendar days after the application has been received.