  We previously obtained a generalization and refinement of results about the
ramification theory of Artin-Schreier extensions of discretely valued fields in
characteristic $p$ with perfect residue fields to the case of fields with more
general valuations and residue fields. As seen in VT16, the "defect" case gives
rise to many interesting complications. In this paper, we present analogous
results for degree $p$ extensions of arbitrary valuation rings in mixed
characteristic $(0,p)$ in a more general setting. More specifically, the only
assumption here is that the base field $K$ is henselian. In particular, these
results are true for defect extensions even if the rank of the valuation is
greater than $1$. A similar method also works in equal characteristic,
generalizing the results of VT16.
