We present a polarization-dependent Raman scattering study of the kagome metal CsV₃Sb₅, focusing on the charge density wave (CDW). We resolve several signatures of strong-coupling between the lattice and the electronic system. First, the energy of the A_1g Γ-point phonon exhibits a discontinuity at the phase transition temperature T_CDW = 95 K. Second, the magnitude of the observed CDW-energy gap 2Δ is compatible with strong coupling. The gap ratio 2Δ/k_BT_CDW exceeds the weak coupling limit of 3.52 by a factor of 6 in both A_1g and E_2g symmetry. Third, the observed A_1g amplitude mode depends weaker on temperature than expected and displays an asymmetric Fano-type line shape, particularly around 70K. We interpret this deviation in terms of strong coupling between the phonon-like amplitude mode and the electronic continuum. To the best of our knowledge such an asymmetry of an amplitude mode has not been observed before.