Suppose that $P \subset M$ is a two-sided real projective plane. Let $M'$ be the orientation double cover of $M$. Let $S$ be the two-sphere in $M'$ that double covers $P$.
Exercise A: $P$ is non-separating in $M$ if and only if $S$ is non-separating in $M'$.
Exercise B: If $S$ separates $M'$, then neither component of $M' - S$ is a three-ball.
We deduce, in both the separating and non-separating cases, that $\pi_2(M')$ is non-trivial; thus $\pi_2(M)$ is non-trivial.