Tame proper base change for discretely ringed adic spaces

We consider a proper morphism $X\to S$ and a locally closed immersion $S^{\mathrm{\prime }}\to S$ of discretely ringed adic spaces and prove proper base change for the tame topology in this setting. More precisely, we show that for an abelian $p$-torsion sheaf ($p=cha{r}^{+}(S)$) on the tame site of $X$ that the base change homomorphism for the derived pushforward along $X\to S$ with the pullback along $S^{\mathrm{\prime }}\to S$ is an isomorphism.