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Radio emission at cm wavelengths from highly star-forming galaxies, such asSMGs, is dominated by synchrotron radiation arising from supernova activity.Using deep, high-resolution ($1\sigma=2.3$ $\mu$Jy beam$^{-1}$; $0.75^{"}$) cmradio-continuum observations taken by the VLA-COSMOS 3 GHz Large Project, westudied the radio-emitting sizes of a flux-limited sample of SMGs in the COSMOSfield. Of the 39 SMGs studied here, 3 GHz emission was detected towards 18 ofthem ($\sim46\pm11\%$) with S/N ratios in the range of ${\rm S/N=4.2-37.4}$.Using 2D elliptical Gaussian fits, we derived a median deconvolved major axisFWHM size of $0.54^{"}\pm 0.11^{"}$ for our 18 SMGs detected at 3 GHz. For the15 SMGs with known redshift we derived a median linear major axis FWHM of$4.2\pm0.9$ kpc. No clear correlation was found between the radio-emitting sizeand the 3 GHz or submm flux density, or the redshift of the SMG. However, thereis a hint of larger radio sizes at $z\sim2.5-5$ compared to lower redshifts.The sizes we derived are consistent with previous SMG sizes measured at 1.4 GHzand in mid-$J$ CO emission, but significantly larger than those seen in the(sub)mm continuum emission. One possible scenario is that SMGs have i) anextended gas component with a low dust temperature, and which can be traced bylow- to mid-$J$ CO line emission and radio continuum emission, and ii) awarmer, compact starburst region giving rise to the high-$J$ line emission ofCO, which could dominate the dust continuum size measurements. Because of therapid cooling of CR electrons in dense starburst galaxies ($\sim10^4-10^5$ yr),the more extended synchrotron radio-emitting size being a result of CRdiffusion seems unlikely. Instead, if SMGs are driven by galaxy mergers theradio synchrotron emission might arise from more extended magnetised ISM aroundthe starburst region.

(Sub)millimetre interferometric imaging of a sample of COSMOS/AzTEC submillimetre galaxies - II. The spatial extent of the radio-emitting regions