It is often said, with varying degrees of rigor or enthusiasm, that every rigid space (say over $\mathbb{C}_p$) has a pro-etale cover which is 'topologically trivial' in some sense. For example, this is hinted at (although never said directly) in section 5.7 of Jared Weinstein's *Reciprocity laws and Galois representations: recent breakthroughs* (pdf).

I was wondering if someone could answer the following questions which, while perhaps naive, are of great interest to me.

If this is too many questions I can break it up into multiple posts.

1) Let $X$ be the `perfectoid closed unit disk' given by $\text{Spa}(\mathbb{C}_p\langle T^{\frac{1}{p^\infty}}\rangle,\mathcal{O}_{\mathbb{C}_p}\langle T^{\frac{1}{p^\infty}}\rangle)$. Is it true that $X-\{0\}$ is 'simply connected' in the sense that $\pi_1^{\acute{e}\text{t}}(X-\{0\})=0$? One might imagine that this is something like a pro-etale universal cover of the usual punctured closed unit disk over $\mathbb{C}_p$ since it's at least $\sim$ (in the language of Scholze) to an inverse limit of the finite etale covers $x\mapsto x^{p^n}$ of the punctured closed disk over $\mathbb{C}_p$. I suspect not since covers of the disk (like Artin-Schreier covers) exist. So, is there a 'universal cover' in this case?

EDIT: As mentioned below, I was being hasty and should have said something like 'the maximal $p$-quotient of $\pi_1^{\acute{e}\text{t}}$ is zero.

2) If $X$ is any rigid space over $K$ (a $p$-adic field, or perhaps $\widehat{\overline{K}}$ for a $p$-adic field) then is there a 'universal cover' of $X$. The precise definition of this is open for me--is there a particularly good notion, and if so, when does it exist. For example, if there is a pro-etale cover $\{U_i\}$ of $X$ and an adic space $\widetilde{X}$ such that $\widetilde{X}\sim \varprojlim U_i$ and $\widetilde{X}$ is 'simply connected' (i.e. that $\pi_1^{\acute{e}{t}}(\widetilde{X})=0$ or, perhaps even better, $\pi_1^{\text{pro}\acute{e}\text{t}}(\widetilde{X})=0$).

EDIT: As mentioned below, I was being hasty and should have said something like 'the maximal $p$-quotient of $\pi_1^{\acute{e}\text{t}}$ is zero and similarly for the proetale fundamental group.

3) I have the sense that perfectoid spaces have 'simpler geometry' (for example, I think this can be seen by the almost zeroness of their cohomology with values in $\mathcal{O}_X^+$ and how this relates to their cohomology with values in $\mathbb{F}_p$ by the AS sequence), but I don't know a precise statement of this. Namely, how is the etale topology of a perfectoid space simpler than, say, a general rigid variety?

4) It is sometimes said that one can use perfectoid geometry to try and compute things like the etale cohomology of some rigid variety by computing the Cech cohomology of some pro-etale perfectoid cover. What is the precise statement of this? I think that one can compute the cohomology of $\mathcal{O}_X^+$ almost (in the technical sense) from a pro-etale perfectoid cover. Does one then try to compute cohomology with coefficients in $\mathbb{F}_p$ almost-ly by using the AS sequence again?

Any answers to any of these questions would be greatly appreciate--as well as any other insight someone might want to add.

Thanks!

EDIT: I guess I should add that one probably doesn't expect universal covers in terms of 'pure topological trivialness' (or, rather, pure topological trivialness in-so far as $\pi_1$ or other '1-dimensional topology' is concerned) but rather in the sort of topological trivialness (again in degree $1$) concerned with $p$-torsion or pro-$p$-torsion coefficients.

So, as Will Sawin mentions below, you can still likely make covers with prime-to-$p$ degree of the 'universal cover'.

In summary, perhaps '$p$-universal cover' is better...