Electronic Raman scattering has been measured at low temperatures from Li-doped CdS, where Li is incorporated as a shallow acceptor. The spectra are interpreted in terms of a LO-phonon mode bound to an impurity site, and in terms of the electronic transitions between bound hole states and the phonon sidebands of these transitions. A strong Fano-type effect accompanied by a low-energy shift is observed in the LO-phonon sideband of the $1\stackrel{\ensuremath{\rightarrow}}{s}2s$ transition due to its overlap with a continuum corresponding to the zero-phonon hole transitions from the $1s$ state to the valance band. A calculation of the self energy in second-order perturbation theory with the Fr\"ohlich interaction shows good agreement with the observed shift. The phonon sideband is also found to resonate with the incident laser energy near the energy gap between the conduction band and the $2s$ acceptor state. This resonance behavior is interpreted in terms of a many-body theory given earlier for the phonon sideband in Li-doped ZnTe, which shows good agreement with experimental results.