Integral model of etale covering

Let $K$ be a $p$-adic field whose residue field $k$ is algebraically closed. Let $X$ be a hyperbolic curve over $K$, what I mean by a curve is a smooth geometrically connected scheme of dimension one over $K$.

Let us further assume that there exists a flat proper morphism $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$ such that $\overline{\mathcal{X}}_K\times_K\overline{K}$ is the smooth completion of $X\times_K\overline{K}$, where $\overline{\mathcal{X}}_K$ is the generic fibre of $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$ as usual.

Now, here is the question:

Q. Let $\overline{\mathcal{X}}_k$ be the special fibre of $\overline{\mathcal{X}}\to\mathrm{Spec}\mathcal{O}_K$.Then, does any finite etale covering of $\overline{\mathcal{X}}_k$ lift to a finite etale covering of $\overline{\mathcal{X}}$?

• With your definition, $\mathbb{P}^1_K$ is a hyperbolic curve. That's probably not what you want. In any case, you are asking whether finite etale covers of a curve over $\overline{\mathbb{F}_p}$ lift. With your definitions, a simple counterexample of non-liftable covers are the Artin-Schreier covers of the affine line. – Ariyan Javanpeykar May 12 '18 at 17:42
• @AriyanJavanpeykar Why is $\mathbb{P}^1_K$ hyperbolic under my definition? I said a "curve" is a smooth geometrically connected scheme of dimension 1, NOT "hyperbolic curve". – User0829 May 13 '18 at 2:39
• You are right. I misread that. My apologies. The answer to your question is still negative though. Choose $X\to \mathbb{A}^1$ finite etale with $X$ hyperbolic. If I recall correctly such covers exist. Now pull-back Artin-Schreier covers to $X$. – Ariyan Javanpeykar May 14 '18 at 18:52
• @AriyanJavanpeykar Thank you. If I restrict myself to finite etale coverings of $\overline{\mathcal{X}}_k$ of $p$-power degree, do I still have negative answer? - the reason for asking this question is, in a paper published in a renowned journal, I found a similar claim stated without any proof (the author said that it is almost trivial fact for experts). – User0829 May 15 '18 at 0:26
• Did you mean to write prime-to-$p$ degree? The answer would then be positive; look up "tame fundamental group" for instance. However, if you really meant $p$-power degree, then the answer is negative, because of the Artin-Schreier covers. – Ariyan Javanpeykar May 15 '18 at 8:21