  We give explicit formulas witnessing IP, \IPn or TP2 in fields with
Artin-Schreier extensions. We use them to control $p$-extensions of mixed
characteristic henselian valued fields, allowing us most notably to generalize
to the \NIPn context one way of Anscombe-Jahnke's classification of NIP
henselian valued fields. As a corollary, we obtain that \NIPn henselian valued
fields with NIP residue field are NIP. We also discuss tameness results for
NTP2 henselian valued fields.
