  We use analytical and N-body methods to examine the survival of wide stellar
binaries against repeated encounters with dark substructures orbiting in the
dark matter haloes of dwarf spheroidal galaxies (dSphs). Our models adopt
cosmologically-motivated conditions wherein dSphs are dark-matter dominated
systems that form hierarchically and orbit about a host galaxy. Our analytical
estimates show that wide binaries are disrupted at a rate that is proportional
to the local density of dark substructures averaged over the life-time of the
binary population. The fact that external tides can efficiently strip dark
substructures from the outskirts of dSphs implies that the present number and
distribution of binaries is strongly coupled with the mass evolution of
individual galaxies. Yet we show that for the range of dynamical masses and
Galactocentric distances spanned by Milky Way dSphs, a truncation in the
separation function at a_max <~ 0.1 pc is expected in all these galaxies. An
exception may be the Sagittarius dSph, which has lost most of is dark matter
envelope to tides and is close to full disruption. Our simulations indicate
that at separations larger than a_max the perturbed binary distribution scales
as dN/da \propto a^{-2.1} independently of the mass and density of
substructures. These results may be used to determine whether the binary
separation function found in dwarf galaxies is compatible with the scale-free
hierarchical picture that envisions the existence of dark substructures in all
galactic haloes. We show that the ACS camera on board of the Hubble telescope
may be able to test this prediction in dSphs at heliocentric distances <100
kpc, even if the binary fraction amounts only 10% of the stellar population.
