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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.

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    $\begingroup$ You might find this article of Weinstein interesting and relevant (math.bu.edu/people/jsweinst/GQpAsFundamentalGroup/…). In this, he shows that the etale fundamental group of the ``punctured perfectoid open ball" quotiented by $\mathbb{Q}_p^{\times}$ is $\text{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$. $\endgroup$
    – Jackson Morrow
    Apr 27, 2021 at 13:48

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