It is widely accepted that phonon-mediated high-temperature superconductivity is impossible at ambient pressure, because of the very large effective masses of polarons or bipolarons at strong electron-phonon coupling. Here we challenge this belief by showing that strongly bound yet very light bipolarons appear for strong Peierls 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 phonon-mediated interaction, which is of "pair hopping" instead of regular density-density type. This unusual effective interaction is bound to have nontrivial consequences for the superconducting state expected to arise at finite carrier concentrations and should favor a large critical temperature.