We present high spatial resolution (0.4 arcsec, kpc) Plateau de Bure Interferometer interferometric data on three ultraluminous infrared galaxies (ULIRGs) at : two submillimetre galaxies (SMGs) and one submillimetre faint star-forming radio galaxy. The three galaxies have been robustly detected in CO rotational transitions, either 12CO (J= 43) or 12CO (J= 32), allowing their sizes and gas masses to be accurately constrained. These are the highest spatial resolution observations observed to date (by a factor of 2) for intermediate-excitation CO emission in ULIRGs. The galaxies appear extended over several resolution elements, having a mean radius of 3.7 kpc. High-resolution (0.3 arcsec) combined Multi-Element Radio-Linked Interferometer Network-Very Large Array observations of their radio continua allow an analysis of the star formation behaviour of these galaxies, on comparable spatial scales to those of the CO observations. This ‘matched beam’ approach sheds light on the spatial distribution of both molecular gas and star formation, and we can therefore calculate accurate star formation rates and gas surface densities: this allows us to place the three systems in the context of a Kennicutt–Schmidt (KS)-style star formation law. We find a difference in size between the CO and radio emission regions, and as such we suggest that using the spatial extent of the CO emission region to estimate the surface density of star formation may lead to error. This size difference also causes the star formation efficiencies within systems to vary by up to a factor of 5. We also find, with our new accurate sizes, that SMGs lie significantly above the KS relation, indicating that stars are formed more efficiently in these extreme systems than in other high-z star-forming galaxies.