We propose a novel mechanism of flat band formation based on the relative biasing of only one sublattice against other sublattices in a honeycomb lattice bilayer. The mechanism allows modification of the band dispersion from parabolic to “Mexican hat”–like through the formation of a flattened band. The mechanism is well applicable for bilayer graphene—both doped and undoped. By angle-resolved photoemission from bilayer graphene on SiC, we demonstrate the possibility of realizing this extremely flattened band (< 2-meV dispersion), which extends two-dimensionally in a k-space area around the Embedded Image point and results in a disk-like constant energy cut. We argue that our two-dimensional flat band model and the experimental results have the potential to contribute to achieving superconductivity of graphene- or graphite-based systems at elevated temperatures.