Context. The CoRoT satellite has recently discovered a hot Jupiter that transits across the disc of a F9 main-sequence star called CoRoT-6 with a period of 8.886 days. Aims: We model the photospheric activity of the star and use the maps of the active regions to study stellar differential rotation and the star-planet interaction. Methods: We apply a maximum entropy spot model to fit the optical modulation as observed by CoRoT during a uninterrupted interval of ~ 140 days. Photospheric active regions are assumed to consist of spots and faculae in a fixed proportion with solar-like contrasts. Results: Individual active regions have lifetimes up to 30-40 days. Most of them form and decay within five active longitudes whose different migration rates are attributed to the stellar differential rotation for which a lower limit of ΔΩ/Ω = 0.12 ± 0.02 is obtained. Several active regions show a maximum of activity at a longitude lagging the subplanetary point by ~ 200° with the probability of a chance occurrence being smaller than 1 percent. Conclusions: Our spot modelling indicates that the photospheric activity of CoRoT-6 could be partially modulated by some kind of star-planet magnetic interaction, while an interaction related to tides is highly unlikely because of the weakness of the tidal force. Based on observations obtained with CoRoT, a space project operated by the French Space Agency, CNES, with partecipation of the Science Programme of ESA, ESTEC/RSSD, Austria, Belgium, Brazil, Germany, and Spain.