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We give explicit formulas witnessing IP, IP
$_{\!n}$, 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 NIP
$_{\!n}$ context one way of Anscombe–Jahnke’s classification of NIP henselian valued fields. As a corollary, we obtain that NIP
$_{\!n}$ henselian valued fields with NIP residue field are NIP. We also discuss tameness results for NTP2 henselian valued fields.
