  We expand the relativistic open bosonic string in powers of $1/c^2$ where $c$
is the speed of light. We perform this expansion to next-to-leading order in
$1/c^2$ and relate our results to known descriptions of non-relativistic open
strings obtained by taking limits. Just as for closed strings the
non-relativistic expansion is well-defined if the open string winds a circle in
the target space. This direction must satisfy Dirichlet boundary conditions. It
is shown that the endpoints of the open string behave as Bargmann particles in
the non-relativistic regime. These open strings end on nrD$p$-branes with $p\le
24$. When these nrD$p$-branes do not fluctuate they correspond to
$(p+1)$-dimensional Newton--Cartan submanifolds of the target space. When we
include fluctuations and worldvolume gauge fields their dynamics is described
by a non-relativistic version of the DBI action whose form we derive from
symmetry considerations. The worldvolume gauge field and scalar field of a
nrD$24$-brane make up the field content of Galilean electrodynamics (GED), and
the effective theory on the nrD$24$-brane is precisely a non-linear version of
GED. We generalise these results to actions for any nrD$p$-brane by demanding
that they have the same target space gauge symmetries that the non-relativistic
open and closed string actions have. Finally, we show that the nrD$p$-brane
action is transverse T-duality covariant. Our results agree with the findings
of Gomis, Yan and Yu in arXiv:2007.01886.
