  We compute the image of the $p$-adic period map for polarized K3 surfaces
with supersingular reduction. This gives rise to a Rapoport-Zink type
uniformization of their moduli space by an explicit open rigid analytic
subvariety of a local Shimura variety of orthogonal type. In contrast to the
case of Rapoport-Zink uniformization of Shimura varieties and in analogy to the
complex case, the uniformizing domain does not carry an action of a $p$-adic
Lie group, but only of a discrete subgroup. We briefly sketch how the same
arguments can be applied to obtain a uniformization for the moduli space of
smooth cubic fourfolds with supersingular reduction.
