Let $K$ a local field, with ring of integer $\mathcal{O}_K$ or $\mathcal{O}$ ,of uniformizer $\pi$, residue field $k$ of caracteristic $p$.
Let $\varphi:\mathbb{P}^1_K\to\mathbb{P}^1_K$ finite separable morphism with tame ramifications: all the order of vanishing are prime to $p$. Let $\Phi:\mathbb{P}^1_{\mathcal{O}}\to\mathbb{P}^1_\mathcal{O}$ an extension of $\varphi$ to $\mathbb{P}^1_\mathcal{O}$ and let $\overline{\varphi}$ the reduction of $\Phi$ in $k$ that is $\overline{\varphi}=\Phi\times_\mathcal{O}\text{Id}_k:\mathbb{P}^1_k\to\mathbb{P}^1_k$. Concretely $\varphi$ is associated with a fraction $\pi^k P/Q$ with $P,Q\in\mathcal{O}[t]$,$\Phi$ is $[\pi^kP:Q]$ or $[P:\pi^{-k}Q]$ (depending of the signe of $k$) and $\overline{\varphi}$ is the reduction of this fraction if it's possible (constant otherwise).
Question: can we tell that if all the ramifications of $\varphi$ are tame and $\overline{\varphi}$ is not constant (ie finite) then $\varphi$ is a separable morphism? If not can you give me an example. If not, is there a criterion for the reduction to stay separable? Thanks!
