# Étale fundamental group of rigid analytification

Let $$X$$ be a quasi-projective variety over a $$p$$-adic field. Denote by $$X^{an}$$ its rigid analytification. Does $$\pi_1^{et}(X)=\pi_1^{et}(X^{an})$$?

Yes, if $$X$$ is proper (by GAGA). Otherwise, this is false for covers of degree divisible by $$p$$. See Example 51 in Ducros's survey "Étale Cohomology of Schemes and Analytic Spaces", though unfortunately he does not give a reference.
For an example of this failure in equal characteristic $$p$$, see section 7.4 in de Jong–van der Put "Etale cohomology of rigid analytic spaces" (Doc. Math. 1995). They construct an example of a $$\mathbf{Z}/p$$-covering of $$\mathbf{A}^{1, \rm an}$$ which is not the analytification of a finite etale covering of the line.
In mixed characteristic, almost the same argument should work if we replace Artin-Schreier coverings with Kummer coverings of degree $$p$$ (i.e. replace equations $$T^p-T=f$$ with $$T^p=f$$), but I didn't check the details.