Context. The star formation rate (SFR) per unit area correlates well with the gas surface density for different types of galaxies. However, this Kennicutt-Schmidt (K-S) law has not yet been examined for a large, homogeneously selected sample of submillimetre galaxies (SMGs), which could provide useful SF implementation information for models of massive galaxy formation and evolution. Aims. We aim at determining the K-S law parameters for the first time for a well-selected, statistical sample of SMGs. Methods. We used the Atacama Large Millimetre/submillimetre Array (ALMA) to conduct a high resolution (0.̋2), 870 μm continuum imaging survey of 40 SMGs, which were initially selected at 1.1 mm in the COSMOS field. We analysed a sample of 32 out of the 40 target SMGs, for which our new ALMA 870 μm data provide information about the spatial extent of dust emission, and all of which have dust-obscured SFR and dust-based gas mass estimates available from our previous study. Results. We divided our sample into equally large subsamples of main-sequence (MS) objects and starbursts (factor of > 3 above the MS), and found their K-S relations to be of the form ΣSFR ∝Σgas0.81±0.01 and ΣSFR ∝Σgas0.84±0.39, respectively. Conclusions. The slightly sub-linear K-S slopes we derived suggest that the SF efficiency (SFE) is nearly constant across the Σgas range probed. Under the assumption of a Galactic CO-to-H2 conversion factor (αCO) for the whole sample, the MS SMGs obey a constant global SFE of about 21% per 100 Myr, while that of starburst SMGs is about 27% per 100 Myr. The corresponding gas depletion times are ~ 480 Myr and 370 Myr. On average, our SMGs have Σgas ≳ 103.9 M⊙ pc-2, which suggests that they are Eddington-limited. This is consistent with the theoretical expectation of a linear K-S relation for such systems. However, size measurements of the CO-emitting regions of SMGs, and the αCO values of SMGs are needed to further constrain their Σgas values.