Let $F \colon \mathbb{R}^m \to \mathbb{R}^n$ be a real analytic function (i.e., each of its component functions is real analytic). Does the set $$Z = \{x \in \mathbb{R}^m : F(x) = 0\}$$ have a smooth point? If not, what is an explicit example of such a set with no smooth points? Assume $Z$ is not empty.
By $x \in Z$ is a smooth point, I mean there is an open set $U \subset \mathbb{R}^m$ containing $x$ such that $Z \cap U$ is a smooth manifold.
[Comments: I know this is true if $F$ is a polynomial map (and so $Z$ is an affine variety).]