Arithmetic $\mathcal{D}$-modules over Laurent series fields: the relative case

Let $k$ be a perfect field of characteristic $p>0$. Within Berthelot's theory of arithmetic $\mathcal{D}$-modules, we construct a $p$-adic formalism of Grothendieck's six operations for quasi-projective schemes over the basis $\mathrm{Spec}\, k[[t]]$.