Kinematically complete experiments have been performed on the two-and three-body exit channels in the reactions 84Kr+166Er and 129Xe+~22Sn at 12.5 MeV/u. Threebody events occur with an unusually high probability. They arise from a fast two-step mechanism where a sequential fission-like process follows a deep inelastic collision with preferentially very large energy losses. Strong Coulomb proximity effects are observed in the three-body final state which, treated quantitatively in Coulomb trajectory calculations, establish a time-scale of 1-10-2~s between the consecutive scission acts. The angular distribution of fission fragments is consistent with an orientation of the fission axis approximately collinear with the axis of the first scission, and the mass distribution of the fission is asymmetric with the heavier mass emitted preferentially opposite to the direction of the third particle. The high fission probability, the short time-scale, the near collinear orientation and the fission mass asymmetry together present consistent evidence for a new phenomenon of non-equilibrium fission.