We present a candidate for the most distant galaxy known to date with a photometric redshift of z = 10.7 (95% confidence limits; with z <9.5 galaxies of known types ruled out at 7.2s). This J-dropout Lyman break galaxy, named MACS0647-JD, was discovered as part of the Cluster Lensing and Supernova survey with Hubble (CLASH). We observe three magnified images of this galaxy due to strong gravitational lensing by the galaxy cluster MACSJ0647.7+7015 at z = 0.591. The images are magnified by factors of ~80, 7, and 2, with the brighter two observed at ~26th magnitude AB (~0.15 µJy) in the WFC3/IR F160W filter (~1.4-1.7 µm) where they are detected at ¿12s. All three images are also confidently detected at ¿6s in F140W (~1.2-1.6 µm), dropping out of detection from 15 lower wavelength Hubble Space Telescope filters (~0.2-1.4 µm), and lacking bright detections in Spitzer/IRAC 3.6 µm and 4.5 µm imaging (~3.2-5.0 µm). We rule out a broad range of possible lower redshift interlopers, including some previously published as high-redshift candidates. Our high-redshift conclusion is more conservative than if we had neglected a Bayesian photometric redshift prior. Given CLASH observations of 17 high-mass clusters to date, our discoveries of MACS0647-JD at z ~ 10.8 and MACS1149-JD at z ~ 9.6 are consistent with a lensed luminosity function extrapolated from lower redshifts. This would suggest that low-luminosity galaxies could have reionized the universe. However, given the significant uncertainties based on only two galaxies, we cannot yet rule out the sharp drop-off in number counts at z ¿ 10 suggested by field searches.