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This work is based on deep multi-band (g, r, i) data from the Fornax Deep Survey with VST. We analyse the surface brightness profiles of the 19 bright ETGs inside the virial radius of the Fornax cluster. The main aim of this work is to identify signatures of accretion onto galaxies by studying the presence of outer stellar halos, and understand their nature and occurrence. Our analysis also provides a new and accurate estimate of the intra-cluster light inside the virial radius of Fornax. We performed multi-component fits to the azimuthally averaged surface brightness profiles available for all sample galaxies. This allows to quantify the relative weight of all components in the galaxy structure that contribute to the total light. In addition, we derived the average g-i colours in each component identified by the fit, as well as the azimuthally averaged g-i colour profiles, to correlate them with the stellar mass of each galaxy and the location inside the cluster. We find that in the most massive and reddest ETGs the fraction of light in, probably accreted, halos is much larger than in the other galaxies. Less-massive galaxies have an accreted mass fraction lower than 30%, bluer colours and reside in the low-density regions of the cluster. Inside the virial radius of the cluster, the total luminosity of the intra-cluster light, compared with the total luminosity of all cluster members, is about 34%. Inside the Fornax cluster there is a clear correlation between the amount of accreted material in the stellar halos of galaxies and the density of the environment in which those galaxies reside. By comparing this quantity with theoretical predictions and previous observational estimates, there is a clear indication that the driving factor for the accretion process is the total stellar mass of the galaxy, in agreement with the hierarchical accretion scenario.