Joint with Franziska Jahnke. Shelah conjectured that a NIP field is either finite, real closed, separably closed, or `like the p-adics'. One meaning of `like the p-adics' is `admits a nontrivial henselian valuation'. Given this version of the conjecture, we study theories of strongly dependent fields, and NIP fields. This builds on recent work of Johnson, Halvi--Hasson, Jahnke--Simon, and others.