Setup:
$k$ is an algebraically closed field.
$\mathcal{O} = k\{t\}$ is the henselization of $k[t]_{(t)}$.
$V \rightarrow \text{Spec}(\mathcal{O})$ is proper and has a section.
$V$ is irreducible, nonsingular, and of dimension $2$.
$X$ is the closed fiber of $V$.
Question:
Why is $\text{ker}(\text{Pic}(V) \rightarrow \text{Pic}(X))$ uniquely divisible by $n$ when $n$ is prime to $\text{char}(k)$?
This is claimed (no proof) in Artin, Grothendieck Topologies, Prop. 4.4.2.