Radio emission at cm wavelengths from highly star-forming galaxies, such as SMGs, is dominated by synchrotron radiation arising from supernova activity. Using deep, high-resolution (\(1\sigma=2.3\) \(\mu\)Jy beam\(^{-1}\); \(0.75^{"}\)) cm radio-continuum observations taken by the VLA-COSMOS 3 GHz Large Project, we studied the radio-emitting sizes of a flux-limited sample of SMGs in the COSMOS field. Of the 39 SMGs studied here, 3 GHz emission was detected towards 18 of them (\(\sim46\pm11\%\)) 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 axis FWHM size of \(0.54^{"}\pm 0.11^{"}\) for our 18 SMGs detected at 3 GHz. For the 15 SMGs with known redshift we derived a median linear major axis FWHM of \(4.2\pm0.9\) 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 \(z\sim2.5-5\) 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. One possible scenario is that SMGs have i) an extended gas component with a low dust temperature, and 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-\(J\) line emission of CO, which could dominate the dust continuum size measurements. Because of the rapid cooling of CR electrons in dense starburst galaxies (\(\sim10^4-10^5\) yr), the more extended synchrotron radio-emitting size being a result of CR diffusion seems unlikely. Instead, if SMGs are driven by galaxy mergers the radio synchrotron emission might arise from more extended magnetised ISM around the starburst region.