We search for galaxies with a strong Balmer break ( Balmer Break Galaxies ; BBGs ) at z \sim 6 over a 0.41 deg ^ { 2 } 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 non-detection in X-ray , optical , far-infrared ( FIR ) , and radio bands . The non-detection in the deep ALMA observations suggests that they are not dusty galaxies but BBGs at z \sim 6 , although contamination from Active Galactic Nuclei ( AGNs ) at z \sim 0 can not be completely ruled out for the moment . Our spectral energy distribution ( SED ) analyses reveal that the BBG candidates at z \sim 6 have stellar masses of \approx 5 \times 10 ^ { 10 } M _ { \odot } dominated by old stellar populations with ages of \ga 700 Myr . Assuming that all the three candidates are real BBGs at z \sim 6 , we estimate the stellar mass density ( SMD ) to be 2.4 ^ { +2.3 } _ { -1.3 } \times 10 ^ { 4 } M _ { \odot } 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 years before the observed epoch of z \sim 6 . We estimate the star-formation rate density ( SFRD ) contributed by progenitors of the BBGs to be 2.4 – 12 \times 10 ^ { -5 } M _ { \odot } yr ^ { -1 } Mpc ^ { -3 } at z > 14 ( 99.7 % confidence range ) . Our result suggests a smooth evolution of the SFRD beyond z = 8 .