We present optical conductivity studies of the type-I clathrate Ba8Ga16Sn30, using a terahertz time-domain spectrometer (0.3-3.0 THz). The lowest-lying spectral peak at 0.72 THz due to the Ba(2) ion's off-center vibration in the oversized cage shows a drastic and anomalous temperature dependence. Below about 100 K, the single broad peak splits into two subpeaks, and with further lowering of the temperature, the spectral shape of this so-called rattling phonon shows non-Boltzmann broadening to the point that the linewidth becomes comparable to the peak frequency. Whereas the initial splitting can be understood by assuming a multiwell anharmonic potential, the strong linewidth broadening toward low temperature cannot, since the Boltzmann factor generally sharpens the low-temperature spectra. The observed behavior suggests strong interaction between the local anharmonic phonons and other excitations.