Is the fixed point set of a self-diffeomorphism of odd order orientable?

Question

Let $M$ be a smooth oriented manifold and let $f \colon M \to M$ be a self-diffeomorphism with $f^p = \text{id}_M$ for some odd $p > 0$. Then $f$ is orientation preserving and it follows from the slice theorem that the fixed point set $$ F = \{ x \in M : f(x) = x\} $$ is a smooth submanifold of $M$. Is $F$ orientable?