  We exhibit the analog of the entropy map for multivariate Gaussian
distributions on local fields. As in the real case, the image of this map lies
in the supermodular cone and it determines the distribution of the valuation
vector. In general, this map can be defined for non-archimedian valued fields
whose valuation group is an additive subgroup of the real line, and it remains
supermodular. We also explicitly compute the image of this map in dimension 3.
