IX OBM

Let $F$ be the face of $P$ closer to $p$ and $p^{\prime }$ be the projection of $p$ in the plane of $F$. Suppose $p^{\prime }$ lies outside of $F$. The line through $p$ and $p^{\prime }$ cuts $P$ in two points $A$ and $B$. Let $A$ be the point between $p$ and $p^{\prime }$. $A$ belongs to a face different from $F$ and the distance from $p$ to $A$ is less than the distance between $p$ and $F$, which contradicts the minimality of $F$. So $p^{\prime }$ lies inside $F$.