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In this paper, we analyse the light distribution in the Hydra I cluster of galaxies to explore their low surface brightness features, measure the intra-cluster light, and address the assembly history of the cluster. For this purpose, we used deep wide-field g- and r-band images obtained with the VST as part of the VEGAS project. The VST mosaic covers ~0.4 times the virial radius around the core of the cluster, which enabled us to map the light distribution down to faint surface brightness levels of mu_g ~ 28 mag/arcsec^2. In this region of the cluster, 44 cluster members are brighter than m_B<16 mag, and the region includes more than 300 dwarf galaxies. Similar to the projected distribution of all cluster members (bright galaxies and dwarfs), we find that the bulk of the galaxy light is concentrated in the cluster core, which also emits in the X-rays, and there are two overdensities: in the north (N) and south-east (SE) with respect to the cluster core. We present the analysis of the light distribution of all the bright cluster members. After removing foreground stars and other objects, we measured the diffuse intra-cluster light and compared its distribution with that of the globular clusters and dwarf galaxies in the cluster. We find that most of the diffuse light low surface brightness features, and signs of possible gravitational interaction between galaxies reside in the core and in the group in the N, while ram-pressure stripping is frequently found to affect galaxies within the SE group. All these features confirm that the mass assembly in this cluster is still ongoing. By combining the projected phase-space with these observed properties, we trace the different stages of the assembly history. We also address the main formation channels for the intra-cluster light detected in the cluster, which contributes ~ 12% to the total luminosity of the cluster., Comment: 18 pages, 13 figures, accepted for publication in A&A