I don't understand what's wrong in the following argument.

Assume that we have a no-arbitrage market where the following products are traded:
 - a risky asset $S$,
 - a risk-free bond $B$,
 - an American put option $P$ with finite maturity $T$ and payoff $K$. Its underlying is $S$.

Now, according to the first fundamental theorem of asset pricing, there exists an equivalent probability measure $\mathbb Q$, under which the two-dimensional process $(\frac{S}{B},\frac{P}{B})$ is a martingale.
But then, $P$ would have the same fair price as an European put option, and this is (as far as I know) false.

Where is the mistake? Have I misunderstood the statement of the theorem?