Abstract The Physics of the Accelerating Universe Survey (PAUS) is an innovative photometric survey with 40 narrow-bands at the William Herschel Telescope (WHT). The narrow-bands are spaced at 100 Å intervals covering the range 4500–8500 Å 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 characterize the photometric redshift performance using PAUS data on the COSMOS field. Comparison to secure spectra from zCOSMOS DR3 shows that PAUS achieves σ68/(1 + $z$) = 0.0037 to iAB < 22.5 for the redshift range 0 < $z$ < 1.2, when selecting the best 50 per cent of the sources based on a photometric redshift quality cut. Furthermore, a higher photo-z precision σ68/(1 + $z$) ∼ 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.