Context. The observed spatial scale of the radio continuum emission from star-forming galaxies can be used to investigate the spatial extent of active star formation, constrain the importance of cosmic-ray transport, and examine the effects of galaxy interactions. Aims. We determine the radio size distribution of a large sample of 152 submillimetre galaxies (SMGs) in the COSMOS field that were pre-selected at 1.1 mm, and later detected with the Atacama Large Millimetre/submillimetre Array (ALMA) in the observed-frame 1.3 mm dust continuum emission at a signal-to-noise ratio (S/N) of ≥5. Methods. We used the deep, subarcsecond- resolution (1σ = 2.3μJy beam-1 ; .̋ 75) centimetre radio continuum observations taken by the Karl G. Jansky Very Large Array (VLA)-COSMOS 3 GHz Large Project. Results. One hundred and fifteen of the 152 target SMGs (76% ± 7%) were found to have a 3 GHz counterpart (≥ 4.2σ), which renders the radio detection rate notably high. The median value of the deconvolved major axis full width at half maximum (FWHM) size at 3 GHz is derived to be 0.̋59 ± 0.̋05 , or 4.6 ± 0.4 kpc in physical units, where the median redshift of the sources is z = 2.23 ± 0.13 (23% are spectroscopic and 77% are photometric values). The radio sizes are roughly log-normally distributed, and they show no evolutionary trend with redshift, or difference between different galaxy morphologies. We also derived the spectral indices between 1.4 and 3 GHz, and 3 GHz brightness temperatures for the sources, and the median values were found to be α1.4 GHz3 GHz = -0.67 (Sν ∝ να) and TB = 12.6 ± 2 K. Three of the target SMGs, which are also detected with the Very Long Baseline Array (VLBA) at 1.4 GHz (AzTEC/C24b, 61, and 77a), show clearly higher brightness temperatures than the typical values, reaching TB(3 GHz) > 104.03 K for AzTEC/C61. Conclusions. The derived median radio spectral index agrees with a value expected for optically thin non-thermal synchrotron radiation, and the low median 3 GHz brightness temperature shows that the observed radio emission is predominantly powered by star formation and supernova activity. However, our results provide a strong indication of the presence of an active galactic nucleus in the VLBA and X-ray-detected SMG AzTEC/C61 (high TB and an inverted radio spectrum). The median radio- emitting size we have derived is ~ 1.5–3 times larger than the typical far-infrared dust-emitting sizes of SMGs, but similar to that of the SMGs’ molecular gas component traced through mid-J line emission of carbon monoxide. The physical conditions of SMGs probably render the diffusion of cosmic-ray electrons inefficient, and hence an unlikely process to lead to the observed extended radio sizes. Instead, our results point towards a scenario where SMGs are driven by galaxy interactions and mergers. Besides triggering vigorous starbursts, galaxy collisions can also pull out the magnetised fluids from the interacting disks, and give rise to a taffy-like synchrotron-emitting bridge. This provides an explanation for the spatially extended radio emission of SMGs, and can also cause a deviation from the well-known infrared-radio correlation owing to an excess radio emission. Nevertheless, further high-resolution observations are required to examine the other potential reasons for the very compact dust-emitting sizes of SMGs, such as the radial dust temperature and metallicity gradients.