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Let $p$ be a prime, let $K$ be a finite extension of $\mathbb{Q}_p$, and let$n$ be a positive integer. We construct equivalences of categories betweencontinuous $p$-adic representations of the $n$-fold product of the absoluteGalois group $G_K$ and $(\varphi, \Gamma)$-modules over one of several rings of$n$-variable power series. The case $n=1$ recovers the original construction ofFontaine and the subsequent refinement by Cherbonnier--Colmez; for general $n$,the case $K = \mathbb{Q}_p$ had been previously treated by the third author. Tohandle general $K$ uniformly, we use a form of Drinfeld's lemma on profinitefundamental groups of products of spaces in characteristic $p$, but forperfectoid spaces instead of schemes. We also construct the multivariateanalogue of the Herr complex to compute Galois cohomology; the case $K =\mathbb{Q}_p$ had been previously treated by Pal and the third author, and wereduce to this case using a form of Shapiro's lemma.

Drinfeld's lemma for perfectoid spaces and overconvergence of multivariate $(\varphi, \Gamma)$-modules