Context. The benchmark open cluster M 67 is known to have solar metallicity and an age similar to that of the Sun. It thus provides us with a great opportunity to study the properties of solar twins, as well as the evolution of Sun-like stars. Aims. Previous spectroscopic studies of M 67 reported possible subtle changes in stellar surface abundances throughout the stellar evolutionary phase, namely the effect of atomic diffusion. In this study we attempt to confirm and quantify more precisely the effect of atomic diffusion, and to explore the level of chemical (in)homogeneity in M 67. Methods. We presented a strictly line-by-line differential chemical abundance analysis of two groups of stars in M 67: three turn-off stars and three subgiants. Stellar atmospheric parameters and elemental abundances were obtained with very high precision using the Keck/HIRES spectra. Results. The subgiants in our sample show negligible abundance variations (≤0.02 dex), which implies that M 67 was born chemically homogeneous. We note that there is a significant abundance difference (~0.1–0.2 dex) between subgiants and turn-off stars, which can be interpreted as the signature of atomic diffusion. Qualitatively stellar models with diffusion agree with the observed abundance results. Some turn-off stars do not follow the general pattern, which suggests that in some cases diffusion can be inhibited, or they might have undergone some sort of mixing event related to planets. Conclusions. Our results pose additional challenges for chemical tagging when using turn-off stars. In particular, the effects of atomic diffusion, which could be as large as 0.1–0.2 dex, must be taken into account in order for chemical tagging to be successfully applied.