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Via the relative fundamental exact sequence of $p$-adic Hodge theory, wedetermine the geometric $p$-adic pro-\'etale cohomology of the Drinfeldsymmetric spaces defined over a $p$-adic field, thus giving an alternativeproof of a theorem of Colmez-Dospinescu-Niziol. Along the way, we describe, interms of differential forms, the geometric pro-\'etale cohomology of thepositive de Rham period sheaf on any connected, paracompact, smoothrigid-analytic variety over a $p$-adic field, and we do it with coefficients. Akey new ingredient is the condensed mathematics recently developed byClausen-Scholze.

On the $p$-adic pro-\'etale cohomology of Drinfeld symmetric spaces