We present the results of a systematic search for galaxies in the redshift range z= 6–9, within the new, deep, near-infrared (Y, J, H) imaging of the Hubble Ultra Deep Field provided by the Wide Field Camera 3 (WFC3) on the Hubble Space Telescope. We have performed full spectral energy distribution fitting to the optical+infrared photometry of all high-redshift galaxy candidates detected at ≥5σ significance in at least one of the WFC3/IR broad-band filters. After careful rejection of contaminants, the result is a sample of 49 galaxies with primary photometric redshift solutions z > 5.9, within the 4.5 arcmin2 field covered by the new near-infrared imaging. Our sample, selected without recourse to specific colour cuts, reselects all but the faintest one of the 16 z850-drops selected by Oesch et al., recovers all five of the Y105-drops reported by Bouwens et al. and adds a further 29 equally plausible galaxy candidates, of which 12 lie beyond z≃ 6.3 and four lie beyond z≃ 7.0. However, we also present confidence intervals on our photometric redshift estimates, including alternative secondary redshift solutions. As a result of this analysis, we caution that acceptable low-redshift (z < 2) solutions exist for 28 out of the 37 galaxies at z > 6.3 and in particular for all eight of the galaxy candidates reported here at z > 7.5. Nevertheless, we note that the very highest redshift candidates appear to be strongly clustered in the field. Based on our photometric redshift analysis, we derive new estimates of the ultraviolet galaxy luminosity function at z≃ 7 and 8. Where our results are most robust, at a characteristic luminosity M1500≃−19.5 (AB), we find that the comoving number density of galaxies declines by a factor of ≃2.5 between z≃ 6 and 7 and by a further factor of ≃2 by z≃ 8. These results suggest that it is difficult for the observed population of high-redshift star-forming galaxies to achieve re-ionization by z≃ 6 without a significant contribution from galaxies well below the detection limits, plus alterations in the escape fraction of ionizing photons and/or continued vigorous star formation at z > 15.