Context. Ultra-short-period planets (USPs) are a unique class of super-Earths with an orbital period of less than a day, and hence they are subject to intense radiation from their host star. These planets cannot retain a primordial H/He atmosphere, and most of them are indeed consistent with being bare rocky cores. A few USPs, however, show evidence for a heavyweight envelope, which could be a water layer resilient to evaporation or a secondary metal-rich atmosphere sustained by outgassing of the molten volcanic surface. Much thus remains to be learned about the nature and formation of USPs. Aims. The prime goal of the present work is to refine the bulk planetary properties of the recently discovered TOI-561 b through the study of its transits and occultations. This is crucial in order to understand the internal structure of this USP and to assess the presence of an atmosphere. Methods. We obtained ultra-precise transit photometry of TOI-561 b with CHEOPS, and performed a joint analysis of these data along with three archival visits from CHEOPS and four TESS sectors. Results. Our analysis of TOI-561 b transit photometry put strong constraints on its properties. In particular, we restrict the uncertainties on the planetary radius at ~2% retrieving R_p = 1.42 ± 0.02 R_⊕. This result informs our internal structure modelling of the planet, which shows that the observations are consistent with a negligible H/He atmosphere; however, other lighter materials are required, in addition to a pure iron core and a silicate mantle, to explain the observed density. We find that this can be explained by the inclusion of a water layer in our model. Additionally, we ran a grid of forward models with a water-enriched atmosphere to explain the transit radius. We searched for variability in the measured R_p/R_★ over time, which could trace changes in the structure of the planetary envelope. However, no temporal variations are recovered within the present data precision. In addition to the transit event, we tentatively detect an occultation signal in the TESS data with an eclipse depth L = 27.40_−11.35^+10.87 ppm. We use models of outgassed atmospheres from the literature to explain this eclipse signal. We find that the thermal emission from the planet can mostly explain the observation. Based on this, we predict that near- to mid-infrared observations with the James Webb Space Telescope should be able to detect silicate species in the atmosphere of the planet. This could also reveal important clues about the planetary interior and help disentangle planet formation and evolution models.