We report on the discovery of a super-Li-rich turnoff (TO) star in the old (12 Gyr), metal-poor ([Fe/H] = –2.1 dex) globular cluster (GC) NGC 6397, based on high-resolution MIKE/Magellan spectra. This star shows an unusually high lithium abundance of A(Li)NLTE = 4.03 ± 0.06 ± 0.14 dex (or, 4.21, accounting for possible contamination from a binary companion) that lies above the canonical Li-plateau by a factor of 100. This is the highest Li enhancement found in a Galactic GC dwarf star to date. We discuss several enhancement mechanisms, but none can unambiguously explain such a high overabundance. The spectrum of the star shows a possible indication of binarity, but its line strengths and chemical element abundance ratios are fully compatible with other TO stars in this GC, seemingly ruling out mass transfer from an asymptotic giant branch companion as origin of the high A(Li). A possible cause is an interaction with a red giant that has undergone cool bottom processing.