A $p$-adic de Rham complex

This is the second in a sequence of three articles exploring the relationship between commutative algebras and $E_\infty$-algebras in characteristic $p$ and mixed characteristic. Given a topological space $X,$ we construct, in a manner analogous to Sullivan's $A_{PL}$-functor, a strictly commutative algebra over $\padic$ which we call the de Rham forms on $X$. We show this complex computes the singular cohomology ring of $X$. We prove that it is quasi-isomorphic as an $E_\infty$-algebra to the Berthelot-Ogus-Deligne \emph{d\'ecalage} of the singular cochains complex with respect to the $p$-adic filtration. We show that one can extract concrete invariants from our model, including Massey products which live in the torsion part of the cohomology. We show that if $X$ is formal then, except at possibly finitely many primes, the $p$-adic de Rham forms on $X$ are also formal. We conclude by showing that the $p$-adic de Rham forms provide, in a certain sense, the "best functorial strictly commutative approximation" to the singular cochains complex.