  Let $\mathcal{K}=(K,v,\ldots)$ be a dp-minimal expansion of a non-trivially
valued field of characteristic $0$ and $\mathcal{F}$ an infinite field
interpretable in $\mathcal{K}$.
  Assume that $\mathcal{K}$ is one of the following: (i) $V$-minimal, (ii)
power bounded $T$-convex, or (iii) $P$-minimal (assuming additionally in (iii)
generic differentiability of definable functions). Then $\mathcal{F}$ is
definably isomorphic to a finite extension $K$ or, in cases (i) and (ii), its
residue field. In particular, every infinite field interpretable in
$\mathbb{Q}_p$ is definably isomorphic to a finite extension of $\mathbb{Q}_p$,
answering a question of Pillay's.
  Using Johnson's work on dp-minimal fields and the machinery developed here,
we conclude that if $\mathcal{K}$ is an infinite dp-minimal pure field then
every field definable in $\mathcal{K}$ is definably isomorphic to a finite
extension of $K$.
  The proof avoids elimination of imaginaries in $\mathcal{K}$ replacing it
with a reduction of the problem to certain distinguished quotients of $K$.

Logic;Commutative Algebra