Since my question was a reference request, I shall be posting the following link to a set of notes on so-called crystalline spaces of proper and separated smooth schemes (which are very similar to de Rham spaces of smooth schemes) and how arithmetic D-modules can be realised as quasi-coherent sheaves on these spaces as an answer. In particular, propositions 7.3 and 7.5 therein seem to be the result that I was looking for.

[R. Gregoric, "Crystalline spaces"][1]


  [1]: https://web.ma.utexas.edu/users/gregoric/CrystalMath.pdf