We investigate the magnetic excitations of the hyperhoneycomb Kitaev magnetβ-Li2IrO3 by means of inelastic Raman scattering. The spectra exhibit the coexistence of a broad scattering continuum and two sharp low-energy peaks at 2.5 and 3 meV, with a distinctive polarization dependence. While the continuum is suggestive of fractional quasiparticles emerging from a proximate quantum spin liquid phase, the sharp peaks provide the first experimental signature of the “non-Loudon- Fleury” one-magnon scattering processes proposed recently. The corresponding microscopic mechanism is similar to the one leading to the symmetric off-diagonal exchange interaction(because it involves a combination of both direct and ligand-mediated exchange paths) but is otherwise completely unexpected within the traditional London-Fleury theory of Raman scattering. The present experimental verification, therefore, calls for a drastic reevaluation of Raman scattering in similar systems with strong spin-orbit coupling and multiple exchange paths.