Radio emission at centimetre wavelengths from highly star-forming galaxies, like submillimetre galaxies (SMGs), is dominated by synchrotron radiation arising from supernova activity. Hence, radio continuum imaging has the potential to determine the spatial extent of star formation in these types of galaxies. Using deep, high-resolution (1σ = 2.3 μJy beam^(-1); 0".75 75) centimetre radio-continuum observations taken by the Karl G. Jansky Very Large Array (VLA)-COSMOS 3 GHz Large Project, we studied the radio-emitting sizes of a flux-limited sample of SMGs in the COSMOS field. The target SMGs were originally discovered in a 1.1 mm continuum survey carried out with the AzTEC bolometer, and followed up with higher resolution interferometric (sub)millimetre continuum observations. Of the 39 SMGs studied here, 3 GHz emission was detected towards 18 of them (~46 ± 11%) with signal-to-noise ratios in the range of S/N = 4.2–37.4. Towards four SMGs (AzTEC2, 5, 8, and 11), we detected two separate 3 GHz sources with projected separations of ~1''̣5–6''̣6, but they might be physically related in only one or two cases (AzTEC2 and 11). Using two-dimensional elliptical Gaussian fits, we derived a median deconvolved major axis FWHM size of 0''̣54±0''̣11 for our 18 SMGs detected at 3 GHz. For the 15 SMGs with known redshift we derived a median linear major axis FWHM of 4.2 ± 0.9 kpc. No clear correlation was found between the radio-emitting size and the 3 GHz or submm flux density, or the redshift of the SMG. However, there is a hint of larger radio sizes at z ~ 2.5–5 compared to lower redshifts. The sizes we derived are consistent with previous SMG sizes measured at 1.4 GHz and in mid-J CO emission, but significantly larger than those seen in the (sub)mm continuum emission (typically probing the rest-frame far-infrared with median FWHM sizes of only ~1.5–2.5 kpc). One possible scenario is that SMGs have i) an extended gas component with a low dust temperature, which can be traced by low- to mid-J CO line emission and radio continuum emission; and ii) a warmer, compact starburst region giving rise to the high-excitation line emission of CO, which could dominate the dust continuum size measurements. Because of the rapid cooling of cosmic-ray electrons in dense starburst galaxies (~10^4–10^5 yr), the more extended synchrotron radio-emitting size being a result of cosmic-ray diffusion seems unlikely. Instead, if SMGs are driven by galaxy mergers – a process where the galactic magnetic fields can be pulled out to larger spatial scales – the radio synchrotron emission might arise from more extended magnetised interstellar medium around the starburst region.