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"

The following link to a set of notes seems to contain I was looking for. It's about so-called crystalline spaces associated to 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, much like how D-modules in characteristic $0$ can be thought of as quasi-coherent sheaves on de Rham spaces (I'm brushing a lot of technicalities under the rug here). 

In particular, propositions 7.3 and 7.5 seem to be the result that I was looking for.

R. Gregoric, "Crystalline spaces"