We investigate the magnetic excitations of the hyperhoneycomb Kitaev magnet β-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 [Yang et al., Phys. Rev. B 104, 144412 (2021)2469-995010.1103/PhysRevB.104.144412]. The corresponding microscopic mechanism is similar to the one leading to the symmetric off-diagonal exchange interaction Γ (because it involves a combination of both direct and ligand-mediated exchange paths) but is otherwise completely unexpected within the traditional Loudon-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.