P-adic Borel extension for Shimura varieties and period images

In 1972, Borel proved that every holomorphic map from a product of punctured unit disks to a complex Shimura variety extends to a map from a product of disks to its Baily--Borel compactification.  Recently, Oswal--Shankar--Zhu and Patel proved the corresponding p-adic statement over discretely valued fields in the abelian-type case using Rapoport--Zink uniformizations.  I will discuss work in progress with Oswal, Shankar, and Yao extending this result to all Shimura varieties for large primes p by instead relying on techniques from p-adic Hodge theory.  I will also describe how the same proof works in the context of geometric period images, at least under a good reduction hypothesis.