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

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 between continuous $p$-adic representations of the $n$-fold product of the absolute Galois 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 of Fontaine and the subsequent refinement by Cherbonnier--Colmez; for general $n$, the case $K = \mathbb{Q}_p$ had been previously treated by the third author. To handle general $K$ uniformly, we use a form of Drinfeld's lemma on profinite fundamental groups of products of spaces in characteristic $p$, but for perfectoid spaces instead of schemes. We also construct the multivariate analogue 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 we reduce to this case using a form of Shapiro's lemma.