p -adic Borel extension for local Shimura varieties

Abstract We show that the moduli spaces of Scholze’s p -adic shtukas with framing satisfy a p -adic rigid analytic version of Borel’s extension theorem. In particular, this holds for local Shimura varieties, for all local Shimura data $(G, [b],\{\mu \})$ , even for exceptional groups G , and extends work of Oswal-Shankar-Zhu-Patel who proved a p -adic Borel extension property for Rapoport-Zink spaces. As a corollary, we deduce that all these spaces satisfy a p -adic rigid analytic version of Brody hyperbolicity.