  The Coma cluster of galaxies hosts the brightest radio halo known and has
therefore been the target of numerous searches for associated inverse Compton
(IC) emission, particularly at hard X-ray energies where the IC signal must
eventually dominate over thermal emission. The most recent search with the
Suzaku Hard X-ray Detector (HXD) failed to confirm previous IC detections with
RXTE and BeppoSAX, instead setting an upper limit 2.5 times below their
nonthermal flux. However, this discrepancy can be resolved if the IC emission
is very extended, beyond the scale of the cluster radio halo. Using
reconstructed sky images from the 58-month Swift BAT all sky survey, the
feasibility of such a solution is investigated. Building on Renaud et al., we
test and implement a method for extracting the fluxes of extended sources,
assuming specified spatial distributions. BAT spectra are jointly fit with an
XMM-Newton EPIC-pn spectrum derived from mosaic observations. We find no
evidence for large-scale IC emission at the level expected from the previously
detected nonthermal fluxes. For all nonthermal spatial distributions
considered, which span the gamut of physically reasonable IC models, we
determine upper limits for which the largest (most conservative) limit is
<4.2x10^{-12} erg/s/cm^2 (20-80 keV), which corresponds to a lower limit on the
magnetic field B>0.2uG. A nominal flux upper limit of <2.7x10^{-12} erg/s/cm^2,
with corresponding B>0.25uG, is derived for the most probable IC distribution
given the size of the radio halo and likely magnetic field radial profile.
