Tame proper base change for discretely ringed adic spaces

We consider a proper morphism $X \to S$ and a locally closed immersion $S' \to S$ 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 ($p = char^+(S)$) on the tame site of $X$ that the base change homomorphism for the derived pushforward along $X \to S$ with the pullback along $S' \to S$ is an isomorphism.