The PAU Survey (PAUS) is an innovative photometric survey with 40 narrow bands at the William Herschel Telescope (WHT). The narrow bands are spaced at 100AA intervals covering the range 4500AA to 8500AA and, in combination with standard broad bands, enable excellent redshift precision. This paper describes the technique, galaxy templates and additional photometric calibration used to determine early photometric redshifts from PAUS. Using BCNz2, a new photometric redshift code developed for this purpose, we characterise the photometric redshift performance using PAUS data on the COSMOS field. Comparison to secure spectra from zCOSMOS DR3 shows that PAUS achieves $sigma_{68} /(1+z) = 0.0037$ to $i_{mathrm{AB}} < 22.5$ when selecting the best 50% of the sources based on a photometric redshift quality cut. Furthermore, a higher photo-z precision ($sigma_{68}/(1+z) sim 0.001$) is obtained for a bright and high quality selection, which is driven by the identification of emission lines. We conclude that PAUS meets its design goals, opening up a hitherto uncharted regime of deep, wide, and dense galaxy survey with precise redshifts that will provide unique insights into the formation, evolution and clustering of galaxies, as well as their intrinsic alignments.
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