  Let $K$ be a non-Archimedean valued field with valuation ring $R$. Let
$C_\eta$ be a $K$-curve with compact type reduction, so its Jacobian $J_\eta$
extends to an abelian $R$-scheme $J$. We prove that an Abel-Jacobi map
$\iota\colon C_\eta\to J_\eta$ extends to a morphism $C\to J$, where $C$ is a
compact-type $R$-model of $J$, and we show this is a closed immersion when the
special fiber of $C$ has no rational components. To do so, we apply a
rigid-analytic "fiberwise" criterion for a finite morphism to extend to
integral models, and geometric results of Bosch and L\"utkebohmert on the
analytic structure of $J_\eta$.
