Via the relative fundamental exact sequence of p-adic Hodge theory, we determine the geometric p-adic pro-étale cohomology of the Drinfeld symmetric spaces defined over a p-adic field, thus giving an alternative proof of a theorem of Colmez-Dospinescu-Nizio l. Along the way, we describe, in terms of differential forms, the geometric pro-étale cohomology of the positive de Rham period sheaf on any connected, paracompact, smooth rigid-analytic variety over a p-adic field, and we do it with coefficients. A key new ingredient is the condensed mathematics recently developed by Clausen-Scholze.