We measure the stellar mass function (SMF) and stellar mass density of galaxies in the COSMOS field up to z ~ 6. We select them in the near-IR bands of the COSMOS2015 catalogue, which includes ultra-deep photometry from UltraVISTA-DR2, SPLASH, and Subaru/Hyper Suprime-Cam. At z> 2.5 we use new precise photometric redshifts with error σz = 0.03(1 + z) and an outlier fraction of 12%, estimated by means of the unique spectroscopic sample of COSMOS (~100 000 spectroscopic measurements in total, more than one thousand having robust zspec> 2.5). The increased exposure time in the DR2, along with our panchromatic detection strategy, allow us to improve the completeness at high z with respect to previous UltraVISTA catalogues (e.g. our sample is >75% complete at 1010 ℳ⊙ and z = 5). We also identify passive galaxies through a robust colour–colour selection, extending their SMF estimate up to z = 4. Our work provides a comprehensive view of galaxy-stellar-mass assembly between z = 0.1 and 6, for the first time using consistent estimates across the entire redshift range. We fit these measurements with a Schechter function, correcting for Eddington bias. We compare the SMF fit with the halo mass function predicted from ΛCDM simulations, finding that at z> 3 both functions decline with a similar slope in thehigh-mass end. This feature could be explained assuming that mechanisms quenching star formation in massive haloes become less effective at high redshifts; however further work needs to be done to confirm this scenario. Concerning the SMF low-mass end, it shows a progressive steepening as it moves towards higher redshifts, with α decreasing from -1.47+0.02-0.02 at z ≃ 0.1 to -2.11+0.30-0.13 at z ≃ 5. This slope depends on the characterisation of the observational uncertainties, which is crucial to properly remove the Eddington bias. We show that there is currently no consensus on the method to quantify such errors: different error models result in different best-fit Schechter parameters. Key words: galaxies: evolution / galaxies: luminosity function, mass function / galaxies: statistics / galaxies: high-redshift⋆ Based on data products from observations made with ESO Telescopes at the La Silla Paranal Observatory under ESO programme ID 179.A-2005 and on data products produced by TERAPIX and the Cambridge Astronomy Survey Unit on behalf of the UltraVISTA consortium (http://ultravista.org/). Based on data produced by the SPLASH team from observations made with the Spitzer Space Telescope (http://splash.caltech.edu).