Shimura varieties are higher dimensional analogues of the modular curve. Borel proved that any holomorphic map from an affine complex algebraic curve to a Shimura variety (with sufficient level structure) must be algebraic. We will discuss a p-adic analogue of this theorem. The talk will mainly focus on the case of the moduli space of principally polarized abelian varieties. This is joint work with Abhishek Oswal and Xinwen Zhu, with an appendix by Anand Patel.