  We present a study of diffuse extragalactic radio emission at $1.75\,$GHz
from part of the ELAIS-S1 field using the Australia Telescope Compact Array.
The resulting mosaic is $2.46\,$deg$^2$, with a roughly constant noise region
of $0.61\,$deg$^2$ used for analysis. The image has a beam size of $150
\times60\,$arcsec and instrumental $\langle\sigma_{\rm n}\rangle= (52\pm5)\,
\mu$Jy beam$^{-1}$. Using point-source models from the ATLAS survey, we
subtract the discrete emission in this field for $S \ge 150\, \mu$Jy
beam$^{-1}$. Comparison of the source-subtracted probability distribution, or
\pd, with the predicted distribution from unsubtracted discrete emission and
noise, yields an excess of $(76 \pm 23) \, \mu$Jy beam$^{-1}$. Taking this as
an upper limit on any extended emission we constrain several models of extended
source counts, assuming $\Omega_{\rm source} \le 2\,$arcmin. The best-fitting
models yield temperatures of the radio background from extended emission of
$T_{\rm b}=(10\pm7) \,$mK, giving an upper limit on the total temperature at
$1.75\,$GHz of $(73\pm10)\,$mK. Further modelling shows that our data are
inconsistent with the reported excess temperature of ARCADE2 to a source-count
limit of $1\, \mu$Jy. Our new data close a loop-hole in the previous
constraints, because of the possibility of extended emission being resolved out
at higher resolution. Additionally, we look at a model of cluster halo emission
and two WIMP dark matter annihilation source-count models, and discuss general
constraints on any predicted counts from such sources. Finally, we report the
derived integral count at $1.4\,$GHz using the deepest discrete count plus our
new extended-emission limits, providing numbers that can be used for planning
future ultra-deep surveys.
