  We determine the Haar measure on the compact $p$-adic special orthogonal
groups of rotations $\mathrm{SO}(d)_p$ in dimension $d=2,3$, by exploiting the
machinery of inverse limits of measure spaces, for every prime $p>2$. We
characterise $\mathrm{SO}(d)_p$ as inverse limits of finite groups, of which we
provide parametrisations and orders, together with an equivalent description
through a multivariable Hensel lifting. Supplying these finite groups with
their normalised counting measures, we get an inverse family of Haar measure
spaces for each $\mathrm{SO}(d)_p$. Finally, we constructively prove the
existence of the so-called inverse limit measure of these inverse families,
which is explicitly computable, and prove that it gives the Haar measure on
$\mathrm{SO}(d)_p$. Our results pave the way towards the study of the
irreducible projective unitary representations of the $p$-adic rotation groups,
with potential applications to the recently proposed $p$-adic quantum
information theory.
