Let $K$ be a complete discrete valued field of unequal characteristic $(0,p)$. The aim of this paper is to describe the semi-stable models for covers $\mathbf P^1_K\to\mathbf P^1_K$ of degree $p$, unramified outside $r\leq p$ points and totally ramified above one of them, under the assumtion that the ramification locus has a particular reduction type (which always occurs if $r\leq 4$). We are principally concerned with the minimal semi-stable models which separate the ramified fibers.