  Dwarf spheroidal (dSph) galaxies are key objects in near-field cosmology,
especially in connection to the study of galaxy formation and evolution at
small scales. In addition, dSphs are optimal targets to investigate the nature
of dark matter. However, while we begin to have deep optical photometric
observations of the stellar population in these objects, little is known so far
about their diffuse emission at any observing frequency, and hence on thermal
and non-thermal plasma possibly residing within dSphs. In this paper, we
present deep radio observations of six local dSphs performed with the Australia
Telescope Compact Array at 16 cm wavelength. We mosaiced a region of radius of
about one degree around three "classical" dSphs, Carina, Fornax, and Sculptor,
and of about half of degree around three "ultra-faint" dSphs, BootesII, Segue2,
and Hercules. The rms noise level is below 0.05 mJy for all the maps. The
restoring beams FWHM ranged from 4.2 x 2.5 arcseconds to 30.0 x 2.1 arcseconds
in the most elongated case. A catalogue including the 1392 sources detected in
the six dSph fields is reported. The main properties of the background sources
are discussed, with positions and fluxes of brightest objects compared with the
FIRST, NVSS, and SUMSS observations of the same fields. The observed population
of radio emitters in these fields is dominated by synchrotron sources. We
compute the associated source number counts at 2 GHz down to fluxes of 0.25
mJy, which prove to be in agreement with AGN count models.
