Semistable reduction of covers of degree p

Let K be a local field of residue characteristic p>0. We explain how to compute the semistable reduction of K-curves Y equipped with a degree-p morphism from Y to the projective line. This includes the reduction at p of superelliptic curves of degree p, but our approach is not limited to Galois covers. We give particular attention to the reduction of plane quartics at p=3, which case is implemented in SageMath. We use the language of non-archimedean analytic geometry in the sense of Berkovich. A key tool is the different function of Cohen, Temkin, and Trushin.