The Banach-Tarski paradox for some subsets of finite-dimensional normed spaces over non-Archimedean valued fields

We show some results related to the classical Banach-Tarski paradox in the setting of finite-dimensional normed spaces over a non-Archimedean valued field K. For instance, all balls and spheres in Kⁿ, and the whole space Kⁿ (for n≥ 2) are paradoxical with respect to certain groups of isometries of Kⁿ. If K is locally compact (e.g., K is the field Qₚ of p-adic numbers for any prime number p), any two bounded subsets of Kⁿ with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of Kⁿ (for n≥ 2).