The classical Banach--Tarski paradox is a counterintuitive result from 1924 concerning the equidecomposability of certain sets in the three-dimensional Euclidean space.
Its ``strong'' form states that, for any bounded sets $A,B\subseteq \mathbb{R}^3$ with nonempty interiors, there exist $n\in\mathbb{N}$, partitions $A=\bigsqcup_{i=1}^{n} A_i$, $B=\bigsqcup_{i=1}^{n} B_i$, and isometries $g_1, \ldots, g_n$ of $\mathbb{R}^3$ such that $g_i(A_i)=B_i$ for all $1\leq i \leq n$.
We then say that $A$ and $B$ are {\em $G$-equidecomposable}, where $G$ is the group of all isometries of $\mathbb{R}^3$.

It follows that any ball in $\mathbb{R}^3$ can be partitioned into a finite number of pieces in such a way that the pieces can be transformed using isometries of $\mathbb{R}^3$ to obtain two balls identical to the initial one, i.e., any ball in $\mathbb{R}^3$ is {\em $G$-paradoxical}. This fact implies the nonexistence of a finitely additive, isometry-invariant measure defined on all subsets of $\mathbb{R}^3$ and normalized on the unit ball.

Until recently, little had been known about related phenomena in contexts other than that of Euclidean spaces, hyperbolic spaces, and some complete real manifolds. This motivates the study of the Banach--Tarski paradox in the setting of finite-dimensional normed spaces over non-Archimedean valued fields.

A {\em valued field} is a field $\mathbb{K}$ endowed with a function $\lvert\cdot\rvert\colon \mathbb{K}\to[0,\infty)$, called a {\em valuation}, satisfying for all $x,y\in \mathbb{K}$ the following conditions:
(1) $\lvert x \rvert=0 \Leftrightarrow x=0$, (2) $\lvert xy \rvert=\lvert x \rvert\lvert y \rvert$, (3) $\lvert x+y \rvert\leq \lvert x \rvert+\lvert y \rvert$.
If, instead of (3), the stronger condition
$\lvert x+y \rvert\leq \max\{\lvert x \rvert,\lvert y \rvert\}$ is satisfied, both the valued field $\mathbb{K}$ and its valuation are called {\em non-Archimedean}. Each valuation induces a metric on $\mathbb{K}$, which is an ultrametric if and only if the valuation is non-Archimedean. It is known that every complete valued field that is not isomorphic (algebraically and topologically) to either $\mathbb{R}$ or $\mathbb{C}$ is non-Archimedean. An important example of a non-Archimedean valued field is the field $\mathbb{Q}_p$ of $p$-adic numbers.

Assume that $\mathbb{K}$ is a non-Archimedean nontrivially valued field and $n\geq 2$. We prove that if $\mathbb{K}^n$ is equipped with a non-Archimedean norm equivalent to the maximum norm, then $\mathbb{K}^n$, all balls, and all spheres in $\mathbb{K}^n$ are paradoxical with respect to the group of affine isometries of $\mathbb{K}^n$ using four pieces. Moreover, if $\mathbb{K}$ is locally compact, then any two bounded subsets of $\mathbb{K}^n$ with nonempty interiors are equidecomposable with respect to the group of affine isometries of $\mathbb{K}^n$.

We also address the one-dimensional case for a complete discretely valued field $\mathbb{K}$.
The group of affine transformations of $\mathbb{K}$ is amenable, hence $\mathbb{K}$ is not paradoxical with respect to affine isometries. Considering the full group of isometries of $\mathbb{K}$ instead, we obtain results similar to those for dimensions $n\ge 2$.

Finally, we formulate a conjecture about measurable equidecomposability of some subsets of $\mathbb{K}^n$ equipped with the Haar measure for a locally compact $\mathbb{K}$. The conjecture is motivated by some recent results on measurable equidecomposability in $\mathbb{R}^n$ with the Lebesgue measure for $n\ge 3$.