  We present a sub-50 pc-scale analysis of the gravitational lens system SDP.81
at redshift 3.042 using Atacama Large Millimetre/submillimetre Array (ALMA)
science verification data. We model both the mass distribution of the
gravitational lensing galaxy and the pixelated surface brightness distribution
of the background source using a novel Bayesian technique that fits the data
directly in visibility space. We find the 1 and 1.3 mm dust emission to be
magnified by a factor of u_tot = 17.6+/-0.4, giving an intrinsic total
star-formation rate of 315+/-60 M_sol/yr and a dust mass of 6.4+/-1.5*10^8
M_sol. The reconstructed dust emission is found to be non-uniform, but composed
of multiple regions that are heated by both diffuse and strongly clumped
star-formation. The highest surface brightness region is a ~1.9*0.7 kpc
disk-like structure, whose small extent is consistent with a potential
size-bias in gravitationally lensed starbursts. Although surrounded by extended
star formation, with a density of 20-30+/-10 M_sol/yr/kpc^2, the disk contains
three compact regions with densities that peak between 120-190+/-20
M_sol/yr/kpc^2. Such star-formation rate densities are below what is expected
for Eddington-limited star-formation by a radiation pressure supported
starburst. There is also a tentative variation in the spectral slope of the
different star-forming regions, which is likely due to a change in the dust
temperature and/or opacity across the source.
