ABSTRACT We present – and make publicly available – accurate and precise photometric redshifts in the ACS footprint from the COSMOS field for objects with iAB ≤ 23. The redshifts are computed using a combination of narrow-band photometry from PAUS, a survey with 40 narrow bands spaced at $100\,\mathring{\rm A}$ intervals covering the range from 4500 to $8500\,\mathring{\rm A}$, and 26 broad, intermediate, and narrow bands covering the UV, visible and near-infrared spectrum from the COSMOS2015 catalogue. We introduce a new method that models the spectral energy distributions as a linear combination of continuum and emission-line templates and computes its Bayes evidence, integrating over the linear combinations. The correlation between the UV luminosity and the O ii line is measured using the 66 available bands with the zCOSMOS spectroscopic sample, and used as a prior which constrains the relative flux between continuum and emission-line templates. The flux ratios between the O ii line and Hα, Hβ and $\mathrm{O\,{\small III}}$ are similarly measured and used to generate the emission-line templates. Comparing to public spectroscopic surveys via the quantity Δz ≡ (zphoto − zspec)/(1 + zspec), we find the photometric redshifts to be more precise than previous estimates, with σ68(Δz) ≈ (0.003, 0.009) for galaxies at magnitude iAB ∼ 18 and iAB ∼ 23, respectively, which is three times and 1.66 times tighter than COSMOS2015. Additionally, we find the redshifts to be very accurate on average, yielding a median of the Δz distribution compatible with |median(Δz)| ≤ 0.001 at all redshifts and magnitudes considered. Both the added PAUS data and new methodology contribute significantly to the improved results. The catalogue produced with the technique presented here is expected to provide a robust redshift calibration for current and future lensing surveys, and allows one to probe galaxy formation physics in an unexplored luminosity-redshift regime, thanks to its combination of depth, completeness, and excellent redshift precision and accuracy.