Two-dimensional (2D) materials consisting of heavy atoms with particular arrangements may host exotic quantum properties. Here, we report a unique 2D semiconducting binary compound, a Sn_{2}Bi atomic layer on Si(111), in which hexagons are formed by bonding Bi with a triangular network of Sn. Because of the unique honeycomb configuration, the heavy elements, and the energy-dependent hybridization between Sn and Bi, 2D Sn_{2}Bi not only shows strong spin-orbit coupling effects but also exhibits high electron-hole asymmetry: Nearly free hole bands and dispersionless flat electron bands coexist in the same system. By tuning the Fermi level, it is possible to preserve both nearly free and strongly localized charge carriers in the same 2D material, which provides an ideal platform for the studies of strongly correlated phenomena and possible applications in nanodevices.