  We report low-temperature (down to $\sim$5 K) Raman signatures of the
recently discovered intercalated honeycomb magnet Ag$_3$LiIr$_2$O$_6$, a
putative Kitaev quantum spin liquid (QSL) candidate. The Kitaev QSL is
predicted to host Majorana fermions as its emergent elementary excitations
through a thermal fractionalization of entangled spins $S = 1/2$. We observe
evidence of this fractionalization in the low-energy magnetic continuum whose
temperature evolution harbours signatures of the predicted Fermi statistics
obeyed by the itinerant Majorana quasiparticles. The magnetic Raman
susceptibility evinces a crossover from the conventional to a Kitaev
paramagnetic state below the temperature of $\sim$80 K. Additionally, the
development of the Fano asymmetry in the low frequency phonon mode and the
enhancement of integrated Raman susceptibilities below the crossover
temperature signifies prominent coupling between the vibrational and Majorana
fermionic excitations.
