The excitations in honeycomb $alpha$-Li$_2$IrO$_3$ have been investigated with high-resolution resonant inelastic x-ray scattering (RIXS) at the O K edge. The low-energy response is dominated by a fully resolved ladder of excitations, which we interpret as being due to multi-phonon processes in the presence of strong electron-phonon coupling (EPC). At higher energies, the orbital excitations are shown to be dressed by phonons. The high quality of the data permits a quantitative test of the analytical model for the RIXS cross-section, which has been proposed to describe EPC in transition metal oxides (TMOs). We find that the magnitude of the EPC is comparable to that found for a range of 3d TMOs. This indicates that EPC may be of equal importance in determining the phenomenology displayed by corresponding 5d based systems.
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