Projecting orthogonal matrix onto a low rank subspace gives orthogonal matrix?

Question

Let $U$ be an $n\times n$ uniformly-random orthogonal matrix, and let $P$ be a projection matrix onto a subspace of rank $k<n$. Say I take the first $m\ll k$ columns of $U$, denoted as $U_1,\cdots,U_m$, and project it onto $P$. Will $PU_1,\cdots,PU_m$ be orthogonal with high probability? Can I bind this probability from below?