The following is likely all obvious to the experts. But since the field looks tricky to an outsider, maybe I may be excused for asking anyway.

I am wondering about basic facts of what would naturally be called the *étale homotopy type* of good versions of non-archimedean smooth rigid analytic spaces. I suspect what I am after works best for smooth Berkovich analytic spaces, but my main interest is not in studying one fixed model for analytic spaces, but in knowing those models which do have a good étale homotopy theory. Therefore in the following I'll say "analytic space" as shorthand for "non-archimedean rigid or Berkovich-style or otherwise analytic space, whichever works best".

By the étale homotopy type of such an analytic space I want to mean the hopefully obvious definition directly analogous to the familiar definition in algebraic geometry.

Now, at least in Berkovich's theory there is already a topological space underlying an analytic space by way of the Berkovich analytic spectrum. My first concrete question is:

*Is the étale homotopy type of a sufficiently well-behaved analytic space equivalent to that of its underlying topological space?*

Berkovich showed that the topological space underlying a smooth Berkovich-analytic space is locally contractible (see here). So the above question has the following sub-question:

*Are polydiscs and/or analytic domains étale contractible?*

*Are analytic spaces locally étale contractible?*

(The last one is really the main point that I am after, since it would imply that the hypercomplete $\infty$-topos over analytic spaces is cohesive, by a similar argument as for complex-analytic geometry. This is something I had been wondering about here on MO a good while back but it really boils down to knowing that smooth analytic spaces are locally étale contractible.)

I imagine that eventually this kind of questions should be particularly interesting when combined with a non-archimedean analytification map from some kind of smooth arithmetic schemes. Then one would naturally wonder if there is a non-archimedean analog of theorem 5.2 in

- Daniel Dugger, Daniel Isaksen,
*Hypercovers in topology*, 2005 (K-theory archive)

which says, in particular, that complex analytification sends hypercovers of smooth schemes over $k \hookrightarrow \mathbb{C}$ to hypercovers of topological spaces/simplicial sets. It seems natural to wonder whether this kind of theorem has a non-archimedean analog. What is known?

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