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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 morphism to extend to integral models, and geometric results of Bosch and Lütkebohmert on the analytic structure of 
$J_\eta $.
