Type Iax supernovae (SNe Iax) represent the largest class of peculiar white dwarf supernovae. The type Iax SN 2012Z in NGC 1309 is the only white dwarf supernova with a detected progenitor system in pre-explosion observations. Deep Hubble Space Telescope (HST) images taken before SN 2012Z show a luminous, blue source that we have interpreted as a helium-star companion (donor) to the exploding white dwarf. We present here late-time HST observations taken similar to 1400 days after the explosion to test this model. We find the SN light curve can empirically be fit by an exponential-decay model in magnitude units. The fitted asymptotic brightness is within 10% of our latest measurements and approximately twice the brightness of the pre-explosion source. The decline of the light curve is too slow to be powered by Co-56 or Co-57 decay: if radioactive decay is the dominate power source, it must be from longer half-life species like Fe-55. Interaction with circumstellar material may contribute to the light curve, as may shock heating of the companion star. Companion-star models underpredict the observed flux in the optical, producing most of their flux in the UV at these epochs. A radioactively heated bound remnant, left after only a partial disruption of the white dwarf, is also capable of producing the observed excess late-time flux. Our analysis suggests that the total ejecta + remnant mass is consistent with the Chandrasekhar mass for a range of SNe Iax.