Characterizing NIP henselian fields

In this paper, we characterize NIP (Not the Independence Property) henselian valued fields modulo the theory of their residue field, both in an algebraic and in a model-theoretic way. Assuming the conjecture that every infinite NIP field is either separably closed, real closed, or admits a nontrivial henselian valuation, this allows us to obtain a characterization of all theories of NIP fields.