If $X$ is a perfectoid space then it has the same étale site as its tilt $X^\flat$. This means that the fundamental groups (suitably defined) of $X$ and $X^\flat$ are isomorphic.

Can you give interesting examples of computation of these fundamental groups (in the spirit of this question)?

For example Jannsen-Wingberg gave a description of the absolute Galois group of $\mathbb{Q}_p$ for $p>2$ but $\mathbb{Q}_p$ of course is not perfectoid.