Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

We show that any normed space $(K^n,\mathrm{\parallel }\cdot \mathrm{\parallel })$, $n\ge 2$, 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 $\mathrm{\parallel }\cdot \mathrm{\parallel }$ is equivalent to the maximum norm. It follows that any finite-dimensional normed space $(X,\mathrm{\parallel }\cdot \mathrm{\parallel })$ with $\dim X\ge 2$ over a complete non-Archimedean nontrivially valued field $(K,\mathrm{\mid }\cdot \mathrm{\mid })$ is paradoxical using four pieces with respect to the group of its affine isometries.