  We define a superspace over a ring $R$ as a functor on a subcategory of the
category of supercommutative $R$-algebras. As an application the notion of a
$p$-adic superspace is introduced and used to give a transparent construction
of the Frobenius map on $p$-adic cohomology of a smooth projective variety over
the ring of $p$-adic integers.
