Paradoxical decompositions of finite-dimensional non-Archimedean normed spaces

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