  We prove that a Kummer surface defined over a complete strictly Henselian
discretely valued field $K$ of residue characteristic different from 2 admits a
strict Kulikov model after finite base change. The Kulikov models we construct
will be schemes, so our results imply that the semistable reduction conjecture
is true for Kummer surfaces in this setup, even in the category of schemes. Our
construction of Kulikov models is closely related to an earlier construction of
K\"unnemann, which produces semistable models of Abelian varieties. It is
well-known that the special fibre of a strict Kulikov model belongs to one of
three types, and we shall prove that the type of the special fibre of a strict
Kulikov model of a Kummer surface and the toric rank of a corresponding Abelian
surface are determined by each other. We also study the relationship between
this invariant and the Galois representation on the second $\ell$-adic
cohomology of the Kummer surface. Finally, we apply our results, together with
earlier work of Halle-Nicaise, to give a proof of the monodromy conjecture for
Kummer surfaces in equal characteristic zero.
