Consider an ultrametic space $X=K^n$ (for the norm $\| x\| =\max_{i=1,n} (\mid {x_1}\mid,\dots,\mid{x_n}\mid)$ where $K$ is an ultrametric field. Let $B(1):=\lbrace x \in X \mid \|x\| \leq 1\rbrace$ be the unit ball. Equip $X$ with Haar measure. Is it possible to partition $B(1)$ into $k$ smaller balls $B_{a_1}(r_1),\dots, B_{a_k}(r_k)$ (where $B_a(r)$ is the closed ball of radius $r$ and center $a$), $k>1$?

Or put in a different way: is there a measure preserving bianalytic map taking $B(1)$ to the union of $B_{a_i}(r_i)$?

Thank you