  In this paper, we study linear spaces of matrices defined over discretely
valued fields and discuss their dimension and minimal rank drops over the
associated residue fields. To this end, we take first steps into the theory of
rank-metric codes over discrete valuation rings by means of skew algebras
derived from Galois extensions of rings. Additionally, we model
projectivizations of rank-metric codes via Mustafin varieties, which we then
employ to give sufficient conditions for a decrease in the dimension.
