Context. The B-W method is used to determine the distance of Cepheids and consists in combining the angular size variations of the star, as derived from infrared surface-brightness relations or interferometry, with its linear size variation, as deduced from visible spectroscopy using the projection factor. The underlying assumption is that the photospheres probed in the infrared and in the visible are located at the same layer in the star whatever the pulsation phase. While many Cepheids have been intensively observed by infrared beam combiners, only a few have been observed in the visible. Aims. This paper is part of a project to observe Cepheids in the visible with interferometry as a counterpart to infrared observations already in hand. Methods. Observations of δ Cep itself were secured with the VEGA/CHARA instrument over the full pulsation cycle of the star. Results. These visible interferometric data are consistent in first approximation with a quasi-hydrostatic model of pulsation surrounded by a static circumstellar environment (CSE) with a size of θCSE = 8.9 ± 3.0 mas and a relative flux contribution of fCSE = 0.07 ± 0.01. A model of visible nebula (a background source filling the field of view of the interferometer) with the same relative flux contribution is also consistent with our data at small spatial frequencies. However, in both cases, we find discrepancies in the squared visibilities at high spatial frequencies (maximum 2σ) with two different regimes over the pulsation cycle of the star, φ = 0.0 − 0.8 and φ = 0.8−1.0. We provide several hypotheses to explain these discrepancies, but more observations and theoretical investigations are necessary before a firm conclusion can be drawn. Conclusions. For the first time we have been able to detect in the visible domain a resolved structure around δ Cep. We have also shown that a simple model cannot explain the observations, and more work will be necessary in the future, both on observations and modelling.