  Let $X$ be a proper smooth toric variety over a perfectoid field of prime
residue characteristic $p$. We study the perfectoid space $\mathcal{X}^{perf}$
which covers $X$ constructed by Scholze, showing that
$\text{Pic}(\mathcal{X}^{perf})$ is canonically isomorphic to
$\text{Pic}(X)[p^{-1}]$. We also compute the cohomology of line bundles on
$\mathcal{X}^{perf}$ and establish analogs of Demazure and Batyrev-Borisov
vanishing. This generalizes the first author's analogous results for
"projectivoid space".
