  We study $D$-dimensional $p$-brane Galilean geometries via the intrinsic
torsion of their adapted connections. These non-Lorentzian geometries are
examples of $G$-structures whose characteristic tensors consist of two
degenerate ``metrics'' of ranks $(p+1)$ and $(D-p-1)$. We carry out the
analysis in two different ways. In one way, inspired by Cartan geometry, we
analyse in detail the space of intrinsic torsions (technically, the cokernel of
a Spencer differential) as a representation of $G$, exhibiting for generic
$(p,D)$ five classes of such geometries, which we then proceed to interpret
geometrically. We show how to re-interpret this classification in terms of
($D-p-2$)-brane Carrollian geometries. The same result is recovered by methods
inspired by similar results in the physics literature: namely by studying how
far an adapted connection can be determined by the characteristic tensors and
by studying which components of the torsion tensor do not depend on the
connection. As an application, we derive a gravity theory with underlying
$p$-brane Galilean geometry as a non-relativistic limit of Einstein--Hilbert
gravity and discuss how it gives a gravitational realisation of some of the
intrinsic torsion constraints found in this paper. Our results also have
implications for gravity theories with an underlying ($D-p-2$)-brane Carrollian
geometry.
