Mod-$p$ Poincaré Duality in $p$-adic analytic geometry

We show Poincaré Duality for ${\bf F}_p$-étale 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 his paper "$p$-adic Hodge theory for rigid-analytic varieties." We prove duality via constructing Faltings' trace map relating Poincaré Duality on the generic fiber to (almost) Grothendieck Duality on the mod-$p$ fiber of a formal model. We also formally deduce Poincaré Duality for $\mathbf{Z}/p^n\mathbf{Z}$, $\mathbf{Z}_p$, and $\mathbf{Q}_p$-coefficients.