Context. The Baade-Wesselink (BW) method, which combines linear and angular diameter variations, is the most common method to determine the distances to pulsating stars. However, the projection factor, p-factor, used to convert radial velocities into pulsation velocities, is still poorly calibrated. This parameter is critical on the use of this technique, and often leads to 5–10% uncertainties on the derived distances.Aims. We focus on empirically measuring the p-factor of a homogeneous sample of 29 LMC and 10 SMC Cepheids for which an accurate average distances were estimated from eclipsing binary systems.Methods. We used the SPIPS algorithm, which is an implementation of the BW technique. Unlike other conventional methods, SPIPS combines all observables, i.e. radial velocities, multi-band photometry and interferometry into a consistent physical modelling to estimate the parameters of the stars. The large number and their redundancy insure its robustness and improves the statistical precision.Results. We successfully estimated the p-factor of several Magellanic Cloud Cepheids. Combined with our previous Galactic results, we find the following P−p relation: −0.08± 0.04(log P−1.18) + 1.24± 0.02. We find no evidence of a metallicity dependent p-factor. We also derive a new calibration of the period-radius relation, log R = 0.684± 0.007(log P−0.517) + 1.489± 0.002, with an intrinsic dispersion of 0.020. We detect an infrared excess for all stars at 3.6 μm and 4.5 μm, which might be the signature of circumstellar dust. We measure a mean offset of Δm3.6 = 0.057 ± 0.006 mag and Δm4.5 = 0.065 ± 0.008 mag.Conclusions. We provide a new P−p relation based on a multi-wavelength fit that can be used for the distance scale calibration from the BW method. The dispersion is due to the LMC and SMC width we took into account because individual Cepheids distances are unknown. The new P−R relation has a small intrinsic dispersion: 4.5% in radius. This precision will allow us to accurately apply the BW method to nearby galaxies. Finally, the infrared excesses we detect again raise the issue of using mid-IR wavelengths to derive period-luminosity relation and to calibrate the Hubble constant. These IR excesses might be the signature of circumstellar dust, and are never taken into account when applying the BW method at those wavelengths. Our measured offsets may give an average bias of ~ 2.8% on the distances derived through mid-IR P−L relations.