For a prime $p$, consider the projective fibered surface $x^p + y^p + z^p = 0$ as a subvariety of $\mathbb{P}^2_{\mathbb{Z}_p}$.
It is smooth over $\mathbb{Q}_p$, but has a nonreduced special fiber. On the other hand, Corollary 9.1.32 in Liu's algebraic geometry book (p388) says the following:
Let $S$ be a Dedekind scheme with function field $K$, and $X\rightarrow S$ be a Noetherian integral projective flat $S$-scheme. Let $P$ be a $K$-rational point, and $D$ the closure of $\{P\}$, and $s\in S$ a closed pint. Then $X_s\cap D$ is reduced to a point $p\in X_s(k(s))$, and $X_s$ is smooth at $p$. In particular, $p$ belongs to a single irreducible component of $X_s$, which is moreover of multiplicity 1 in $X_s$.
How does this make sense in the setting of our surface $x^p + y^p + z^p = 0$? Its special fiber has a single irreducible component of multiplicity $p$, and hence every point on the generic fiber (e.g., $(1,-1,0)$ for $p$ odd) specializes to a nonsmooth point of the special fiber. What am I missing?