This year's Arizona Winter School was on perfectoid spaces, and there were quite a few homotopy theorists in the audience. I'd like to get a "big list" of reasons homotopy theorists might care about perfectoid spaces. Of course, a general answer is that homotopy theorists know by now to pay attention whenever number theorists get excited about something, but I'm looking for specific examples.
Here are a few examples I know of:
$THH$ and $TC$ are intimately related to the $A\Omega$ cohomology theory introduced by Bhatt-Morrow-Scholze. This is my favorite example.
Perfectoid spaces feature heavily in Scholze-Weinstein's paper on moduli of $p$-divisible groups, and homotopy theorists really like $p$-divisible groups. I don't know if there are specific ideas yet on how to apply their work in homotopy theory, though.
I've talked to some people interested in applying them to the study of Lubin-Tate space and the Gross-Hopkins period map.
There's some speculation that it could help with $TAF$.
I don't know much about any of these examples past the first, so please feel free to expand on these examples in addition to providing new ones. I'd also be happy to hear about applications of related things like $p$-adic Hodge theory in general or the pro-étale site, even if they don't explicitly mention perfectoid spaces.