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.