We study winds in high-redshift galaxies driven by a relativistic cosmic ray (proton) component in addition to the hot thermal gas component. Cosmic rays (CRs) are likely to be efficiently generated in supernova shocks inside galaxies. We obtain solutions of such CR-driven free winds in a gravitational potential of the Navarro–Frenk–White form, relevant to galaxies. CRs naturally provide the extra energy and/or momentum input to the system, needed for a transonic wind solution in a gas with adiabatic index γ= 5/3. We show that CRs can effectively drive winds even when the thermal energy of the gas is lost due to radiative cooling. These wind solutions predict an asymptotic wind speed closely related to the circular velocity of the galaxy. Furthermore, the mass outflow rate per unit star formation rate (ηw) is predicted to be ∼0.2–0.5 for massive galaxies, with masses M∼ 1011–1012 M⊙. We show ηw to be inversely proportional to the square of the circular velocity. Magnetic fields at the μG levels are also required in these galaxies to have a significant mass loss. A large ηw for small mass galaxies implies that CR-driven outflows could provide a strong negative feedback to the star formation in dwarf galaxies. Further, our results will also have important implications to the metal enrichment of the intergalactic medium. These conclusions are applicable to the class of free wind models where the source region is confined to be within the sonic point.