  We show that there is a good notion of irreducible sympelectic varieties of
$\mathrm{K3}^{[n]}$-type over an arbitrary field of characteristic zero or $p >
n + 1$. Then we construct mixed characteristic moduli spaces for these
varieties. Our main result is a generalization of Ogus' crystalline Torelli
theorem for supersingular K3 surfaces. For applications, we answer a slight
variant of a question asked by F. Charles on moduli spaces of sheaves on K3
surfaces and give a crystalline Torelli theorem for supersingular cubic
fourfolds.
