Collisionless plasma instabilities operating on the electron time scale can be of importance to explain magnetic field generation; however, their role has been addressed in scenarios where velocity shear is absent. We show that, whenever present, velocity shears must be considered in the electron time scale since electromagnetic perturbations are unstable, both in the parallel (two fluids Kelvin–Helmholtz instability) and in the transverse plane of the flow. Using the two-fluid formalism in the limit of cold plasma, we derive the dispersion relations of the instability in the parallel plane. As the instabilities grow, we also show, through a kinetic model, the development of a dc magnetic field, extending all over the shear region, that reaches a typical value at a saturation of , where β0 and γ0 are respectively the normalized velocity and the Lorentz factor of the plasma flow.