  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 K\"unneth formula in this setting.
