Context. The formation of intracluster light and extended halos around brightest cluster galaxies is closely related to morphological transformation, tidal stripping, and the disruption of galaxies in clusters. Aims. Here we look for observational evidence to characterize these processes, by studying the morphology and kinematics of the diffuse light in the core of the Hydra I cluster. Methods. With V-band surface photometry, we derive the structural parameters (Sersic index n, effective radius R e , b/a, and major axis position angle PA) of the two giant elliptical galaxies NGC 3311 and NGC 3309 in the cluster core. We construct a two-dimensional photometric model, and investigate the diffuse light structures in the residual image after subtracting the two-dimensional model. We also analyze deep long-slit spectra, and establish a link between the structures in the light distribution, the absorption-line kinematics, and the line-of-sight velocity distributions of nearby galaxies and planetary nebulae (PNs). Results. The central galaxy NGC 3311 is surrounded by an extended, symmetric outer halo with n = 10 and an additional, offcentered envelope whose centroid is shifted by about 50 to the north-east. Its luminosity L V = 1.2 × 10 10 (±6.0 × 10 8 ) L corresponds to ∼50% of the luminosity of the symmetric halo in the same region (∼15% of its entire luminosity). The velocity dispersion of the halo rises to cluster core values, ∼400−500 km s −1 , for R > 20 . On the basis of measured PN velocities, at least part of the offcentered envelope consists of high-velocity accreted stars. We also discover two tidal streams in the cluster center, emerging from the dwarf galaxy HCC 026 and the S0 galaxy HCC 007. The HCC 026 stream is redshifted by ∼1200 km s −1 with respect to NGC 3311 (V N3311 3800 km s −1 ), as for HCC 026 itself, a fraction of PNs in the off-centered envelope, and several other dwarf galaxies nearby. The stars in one of the HCC 026 tails are known to be consistent with the low-metallicity population of HCC 026, and our photometry shows that this galaxy is almost dissolved by the tidal field. The tidal stream around HCC 007 extends over at least ∼110 kpc, is fairly thick, and is brighter on the side of the asymmetric outer halo of NGC 3311, which it may join. Its luminosity is several 10 9 L , similar to the luminosity of the stripped-down galaxy HCC 007. The redshift of the stream is determined from a few PN velocities and is similar to those of both HCC 007 and HCC 026.Conclusions. An entire group of small galaxies is currently falling through the core of the Hydra I cluster; these galaxies have already been partially dissolved by the strong tidal field. Their light is being added to the outer halo and intracluster light around the cD galaxy NGC 3311. The Hydra I cluster provides a vivid example of the morphological transformation and tidal dissolution of galaxies in clusters.