The very nearby Type Ia supernova 2014J in M82 offers a rare opportunity to study the physics of thermonuclear supernovae at extremely late phases (greater than or similar to 800 days). Using the Hubble Space Telescope, we obtained 6 epochs of high-precision photometry for SN 2014J from 277 days to 1181 days past the B-band maximum light. The reprocessing of electrons and X-rays emitted by the radioactive decay chain Co-57 -> Fe-57 is needed to explain the significant flattening of both the F606W-band and the pseudo-bolometric light curves. The flattening confirms previous predictions that the late-time evolution of type Ia supernova luminosities requires additional energy input from the decay of Co-57. By assuming the F606W-band luminosity scales with the bolometric luminosity at similar to 500 days after the B-band maximum light, a mass ratio Ni-57/Ni-56 similar to 0.065(-0.005)(+0.004) is required. This mass ratio is roughly similar to 3 times the solar ratio and favors a progenitor white dwarf with a mass near the Chandrasekhar limit. A similar fit using the constructed pseudo-bolometric luminosity gives a mass ratio Ni-57/Ni-56 similar to 0.066(-0.009)(+0.008) . Astrometric tests based on the multi-epoch HST ACS/WFC images reveal no significant circumstellar light echoes in between 0.3 and 100 pc from the supernova.