The Primitive Comparison Theorem in characteristic $p$

We prove an analogue of Scholze’s Primitive Comparison Theorem for proper rigid spaces over an algebraically closed non-archimedean field $K$ of characteristic $p$. This implies a v-topological version of the Primitive Comparison Theorem for proper finite type morphisms $f:X\to Y$ of analytic adic spaces over ${\mathbb{Z}}_p$. We deduce new cases of the Proper Base Change Theorem for $p$-torsion coefficients and the Künneth formula in this setting.