We search for galaxies with a strong Balmer break (Balmer break galaxies; BBGs) at z ~ 6 over a 0.41 deg² effective area in the COSMOS field. Based on rich imaging data, including data obtained with the Atacama Large Millimeter/submillimeter Array (ALMA), three candidates are identified by their extremely red K–[3.6] colors, as well as by nondetection in the X-ray, optical, far-infrared, and radio bands. The nondetection in the deep ALMA observations suggests that they are not dusty galaxies but BBGs at z ~ 6, although contamination from active galactic nuclei at z ~ 0 cannot be completely ruled out for the moment. Our spectral energy distribution analyses reveal that the BBG candidates at z ~ 6 have stellar masses of ≈5 × 10¹⁰ M_⊙ dominated by old stellar populations with ages of ≳ 700 Myr. Assuming that all three candidates are real BBGs at z ~ 6, we estimate the stellar mass density to be 2.4_(-1.3)^(+2.3) x {10⁴ M_⊙ Mpc⁻³. This is consistent with an extrapolation from the lower-redshift measurements. The onset of star formation in the three BBG candidates is expected to be several hundred million yr before the observed epoch of z ~ 6. We estimate the star formation rate density (SFRD) contributed by progenitors of the BBGs to be 2.4–12 × 10⁻⁵ M_⊙ yr⁻¹ Mpc⁻³ at z > 14 (99.7% confidence range). Our result suggests a smooth evolution of the SFRD beyond z = 8
