A very exotic process of β-delayed fission of Tl180 is studied in detail by using resonant laser ionization with subsequent mass separation at ISOLDE (CERN). In contrast to common expectations, the fission-fragment mass distribution of the post-β-decay daughter nucleus Hg180 (N/Z=1.25) is asymmetric. This asymmetry is more surprising since a mass-symmetric split of this extremely neutron-deficient nucleus would lead to two Zr90 fragments, with magic N=50 and semimagic Z=40. This is a new type of asymmetric fission, not caused by large shell effects related to fragment magic proton and neutron numbers, as observed in the actinide region. The newly measured branching ratio for β-delayed fission of Tl180 is 3.6(7)×10⊃-3%, approximately 2 orders of magnitude larger than in an earlier study.