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 \(\mathbb {K}\) . For instance, all balls and spheres in \(\mathbb {K}^n\) , and the whole space \(\mathbb {K}^n\) (for \(n\ge 2\) ) are paradoxical with respect to certain groups of isometries of \(\mathbb {K}^n\) . If \(\mathbb {K}\) is locally compact (e.g., \(\mathbb {K}\) is the field \(\mathbb {Q}_p\) of p-adic numbers for any prime number p), any two bounded subsets of \(\mathbb {K}^n\) with nonempty interiors are equidecomposable (and paradoxical) with respect to a certain group of isometries of \(\mathbb {K}^n\) (for \(n\ge 2\) ).
