  Besicovitch showed that a compact set in $\mathbb{R}^n$ which contains a unit
line segment in every direction can have measure $0$. These constructions also
work over other metric spaces like the $p$-adics and profinite integers. It is
conjectured that it is impossible to construct sets with measure $0$ which
contain a unit $2$-disk in every direction in $\mathbb{R}^n$. We prove that
over the $p$-adics and profinite integers any set which contains a 2-flat in
every direction must have positive measure. The main ingredients are maximal
Kakeya estimates for $(\mathbb{Z}/N\mathbb{Z})^n$ proven in [Dha22] and
adapting Fourier analytic arguments of Oberlin [Obe05]. In general, we prove
$L^{n-1}$ to $L^{n-1}$ estimates for the maximal operator corresponding to
$2$-flats.
