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) for an example of a $\mathbf{Z}/p$-covering of $\mathbf{A}^{1, \rm an}_{\mathbf{Q}_p}$ 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.

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.