  We prove that NIP valued fields of positive characteristic are henselian.
Furthermore, we partially generalize the known results on dp-minimal fields to
dp-finite fields. We prove a dichotomy: if K is a sufficiently saturated
dp-finite expansion of a field, then either K has finite Morley rank or K has a
non-trivial Aut(K/A)-invariant valuation ring for a small set A. In the
positive characteristic case, we can even demand that the valuation ring is
henselian. Using this, we classify the positive characteristic dp-finite pure
fields.
