We show Poincar\'e Duality for $\mathbf{ F}_p$-\'etale cohomology of a smooth proper rigid-analytic space over a non-archimedean field $K$ of mixed characteristic $(0, p)$. It positively answers the question raised by P. Scholze in [Sch13a]. We prove duality via constructing Faltings' trace map relating Poincar\'e Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincar\'e Duality for $\mathbf{ Z}/p^n\mathbf{ Z}$, $\mathbf{ Z}_p$, and $\mathbf{ Q}_p$-coefficients.
Categories: math.AG math.NT