A new lithium silver stannide, Li17Ag3Sn6, was synthesized from high-temperature reactions of the pure elements in tantalum containers. Its crystal structure, in the space group, P31m, with a = 8.063(3) A, c = 8.509(4) A, Z = 1, features two distinct AgSn-based anionic layers. Defect graphitic layers of Ag2Sn3, with ordered vacancies at one-third of the Ag sites, are alternately stacked with Kagome-like nets of isolated trigonal planar AgSn3 units. Double layers of Li ions are sandwiched between the stacked AgSn-based layers. Theoretical calculations show unusual pi-interactions within both anionic layers, with the trigonal planar [AgSn3]11- units being isoelectronic with CO(3)2-. In addition, the chemical bonding of the layered [Ag2Sn3]6- pi-network features incompletely filled lone-pair Sn states involved in in-plane trefoil aromatic interactions. Transport and magnetic susceptibility measurements on Li17Ag3Sn6 indicate excellent metallic behavior and temperature-independent paramagnetism consistent with results from band structure calculations. The "trefoil" aromaticity, previously postulated for aromatic molecular systems, is finally observed, albeit in a polar intermetallic solid-state structure that lies at the border between metals and nonmetals.