A central goal in the exploration of kagome-based materials is the realization of a flat band that has meV bandwidth and lies close to the Fermi energy. The prevailing assumption is that band flattening originates from destructive hopping processes on the kagome lattice. We perform scanning tunneling microscopy (STM) on the layered kagome ferromagnet Fe₃Sn₂ and show that it indeed hosts a flat band near the Fermi energy, which is manifested as a sharp peak in the differential tunneling conductance. First-principles slab calculations reveal, however, that this band is flattened not by the destructive interference of intralayer hopping, but by interlayer hopping between rhombohedral-stacked kagome planes in Fe₃Sn₂, and is confined to the surface layer. This surface band, forming in the vicinity of the Brillouin zone corners K̄ and K'̄, exhibits rich magnetic-field dependence, including fine structure due to valley-symmetry breaking by rotated Fe moments, as well as a persistent diamagnetic shift associated with orbital magnetic moments, all reproduced by our calculations. Our results, highlighting the crucial role of layer stacking on the band structure of kagome magnets, demonstrate experimentally an alternative mechanism of generating magnetically tunable flat bands in atomically thin volumes of topological magnets.