I already asked this on math.stackexchange.com, but didn't receive an answer.

Suppose $f: X \to Y$ is a (possibly proper) morphism of complex manifolds (resp. smooth varieties) such that all fibers of $f$ are of constant dimension $n = \dim X - \dim Y$. We may also suppose that $f$ has connected fibers, I don't know if this is relevant. Also suppose that $Z \subset Y$ is an analytic subset of codimension $\operatorname{codim}(Z, Y) \geq 2$, such that $f$ is smooth over $f^{-1}(Y \setminus Z)$.

Question: Is it true that the fibers of $f$ are generically reduced?