I already asked this on [math.stackexchange.com](https://math.stackexchange.com/questions/4016777/can-non-reduced-fibers-appear-over-a-subset-of-codimension-geq-2), 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?

**Context:** I encountered this problem when reading the paper [*Characteristic Foliation on the Discriminant Hypersurface of a Holomorphic Lagrangian Fibration*](https://doi.org/10.1353/ajm.0.0062) by J.-M. Hwang and K. Oguiso. There one has $\dim X = 2n =  2 \dim Y$, and they argue as follows:
Suppose $z \in Z$, and let $x \in f^{-1}(z)$ be a point which is a smooth point in the reduction $f^{-1}(z)_{\text{red}}$. Let $W \subset  X$ be a smooth $n$-dimensional submanifold, which is meets $f^{-1}(z)_{\text{red}}$ at $x$ transversally. Then they claim that (possibly after shrinking $Y$), the restricted map $f|_W: W \to Y$ is unramified over $Y \setminus Z$ (I don't know why that is true), and hence by the purity of branched loci, $f$ is unramified over $B$. So in particular, $f|_W^{-1}(z) = f^{-1}(z) \cap W$ is smooth at $x$, so $f^{-1}(z)$ is reduced at $x$.