Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

We show that any normed space (Kⁿ,·), n≥ 2, over a field K equipped with a nontrivial non-Archimedean valuation admits a paradoxical decomposition using four pieces with respect to the group of its affine isometries, provided that the norm · is equivalent to the maximum norm. It follows that any finite-dimensional normed space (X,·) with dimX≥ 2 over a complete non-Archimedean nontrivially valued field (K,·) is paradoxical using four pieces with respect to the group of its affine isometries.