Lattices of minimal covolume have been studied fairly intensively in real Lie groups, particularly in the hyperbolic isometry groups. On the other hand, their $p$-adic analogues only have been determined in (some) lower rank groups. In this talk, we will discuss the higher rank behavior of lattices of minimal covolume in $\mathrm{SL}_n(\mathbb{Q}_p)$. We will briefly introduce their general structure, then use Prasad's volume formula and Borel-Prasad techniques to compute their covolume. This quantity involves a variety of number-theoretical objets, and its understanding gives rise to some number-theoretical questions. As this is work in progress, joint with Alireza Salehi Golsefidy, the scope of the talk will be to give a general overview of the techniques and problems involved, rather than stating precise results.