  We prove that Shimura varieties of abelian type satisfy a $p$-adic
Borel-extension property over discretely valued fields. More precisely, let
$\mathsf{D}$ denote the rigid-analytic closed unit disc and
$\mathsf{D}^{\times} = \mathsf{D} \setminus \{0\}$, let $X$ be a smooth
rigid-analytic variety, and let $S(G,\mathcal{H})_{\mathsf{K}}$ denote a
Shimura variety of abelian type with torsion-free level structure. We prove
every rigid-analytic map defined over a discretely valued $p$-adic field
$\mathsf{D}^{\times} \times X \rightarrow
S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{an}}$ extends to an analytic map
$\mathsf{D} \times X \rightarrow
(S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}})^{\textrm{an}}$, where
$S(G,\mathcal{H})_{\mathsf{K}}^{\textrm{BB}}$ is the Baily-Borel
compactification of $S(G,\mathcal{H})_{\mathsf{K}}$. We also deduce various
applications to algebraicity of analytic maps, degenerations of families of
abeloids, and to $p$-adic notions of hyperbolicity. Along the way, we also
prove an extension result for Rapoport-Zink spaces.
