It is widely accepted that phonon-mediated high-temperature superconductivity is impossible at ambient pressure, because of the very large effective masses of polarons/bipolarons at strong electron-phonon coupling. Here we challenge this belief by showing that strongly bound yet very light bipolarons appear for strong Peierls/Su-Schrieffer-Heeger coupling. These bipolarons also exhibit many other unconventional properties, e.g. at strong coupling there are two low-energy bipolaron bands that are stable against strong Coulomb repulsion. Using numerical simulations and analytical arguments, we show that these properties result from the specific form of the phononmediated interaction, which is of "pair-hopping" instead of regular density-density type. This unusual effective interaction is bound to have non-trivial consequences for the superconducting state expected to arise at finite carrier concentrations, and should favor a large critical temperature.