  A field $K$ in a ring language $\mathcal{L}$ is finitely undecidable if
$\mbox{Cons}(\Sigma)$ is undecidable for every nonempty finite $\Sigma
\subseteq \mbox{Th}(K; \mathcal{L})$. We extend a construction of Ziegler and
(among other results) use a first-order classification of Anscombe and Jahnke
to prove every NIP henselian nontrivially valued field is finitely undecidable.
We conclude (assuming the NIP Fields Conjecture) that every NIP field is
finitely undecidable. This work is drawn from the author's PhD thesis.
