We investigate the physical properties of a sample of six submillimeter galaxies (SMGs) in the COSMOS field, spectroscopically confirmed to lie at redshifts z> 4. While the redshifts for four of these SMGs were previously known, we present here two newly discovered z_(spec)> 4 SMGs. For our analysis we employ the rich (X-ray to radio) COSMOS multi-wavelength datasets. In particular, we use new data from the Giant Meterwave Radio Telescope (GMRT) 325 MHz and 3 GHz Jansky Very Large Array (VLA) to probe the rest-frame 1.4 GHz emission at z = 4, and to estimate the sizes of the star formation regions of these sources, respectively. We find that only oneSMG is clearly resolved at a resolution of 0''̣6 × 0''̣7 at 3 GHz, two may be marginally resolved, while the remaining three SMGs are unresolved at this resolution. Combining this with sizes from high-resolution (sub-)mm observations available in the literature for AzTEC 1 and AzTEC 3 we infer a median radio-emitting size for our z> 4 SMGs of (0''̣63 ± 0''̣12) × (0''̣35 ± 0''̣05) or 4.1 × 2.3 kpc^2 (major × minor axis; assuming z = 4.5) or lower if we take the two marginally resolved SMGs as unresolved. This is consistent with the sizes of star formation regions in lower-redshift SMGs, and local normal galaxies, yet higher than the sizes of star formation regions of local ultraluminous infrared galaxies (ULIRGs). Our SMG sample consists of a fair mix of compact and more clumpy systems with multiple, perhaps merging, components. With an average formation time of ~280 Myr, as derived through modeling of the UV IR spectral energy distributions, the studied SMGs are young systems. The average stellar mass, dust temperature, and IR luminosity we derive are M⋆ ~ 1.4 × 10^(11) M⊙, T_(dust) ~ 43 K, and L_(IR) ~ 1.3 × 10^(13)L⊙, respectively. The average L_(IR) is up to an order of magnitude higher than for SMGs at lower redshifts. Our SMGs follow the correlation between dust temperature and IR luminosity as derived for Herschel-selected 0.1 <z< 2 galaxies. We study the IR-radio correlation for our sources and find a deviation from that derived for z< 3 ULIRGs (⟨ q_(IR) ⟩ = 1.95 ± 0.26 for our sample, compared to q ≈ 2.6 for IR luminous galaxies at z< 2). In summary, we find that the physical properties derived for our z> 4 SMGs put them at the high end of the L_(IR)–T_(dust) distribution of SMGs, and that our SMGs form a morphologically heterogeneous sample. Thus, additional in-depth analyses of large, statistical samples of high-redshift SMGs are needed to fully understand their role in galaxy formation and evolution.