We searched for z > 7 Lyman-break galaxies (LBGs) in the optical-to-mid-infrared Hubble Frontier Field and associated parallel field observations of the strong-lensing cluster MACS J0416-2403. We discovered 22 candidates, of which six lie at z > 9 and one lies at z > 10. Based on the Hubble and Spitzer photometry, all have secure photometric redshifts and a negligible probability of being at lower redshifts, according to their peak probability ratios, R. This substantial increase in the number of known high-redshift galaxies allows a solid determination of the luminosity function at z > 8. The number of high-z candidates in the parallel field is considerably higher than that in the Abell 2744 parallel field. Our candidates have median stellar masses of log(M_*) ~ 8.40^{+0.55}_{-0.31}~Msun, SFRs of ~ 1.6^{+0.5}_{-0.4} Msun yr^-1, and SFR-weighted ages of < 310^{+70}_{-140} Myr. Finally, we are able to put strong constraints on the z = 7,8,9 and 10 luminosity functions. One of the objects in the cluster field is a z ~ 10 candidate, with a magnification of mu ~ 20 +- 13. This object is likely the faintest z ~ 10 object known to date, allowing a first look into the extreme faint-end (L ~ 0.04L*) of the z ~ 10 luminosity function.
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