The manifestation of fractals in soliton dynamics has been observed for the first time. The experiment utilized self-generated spin-wave envelope solitons in a magnetic film based active feedback ring. At high ring gain, the soliton that circulates in the ring breathes in a fractal pattern. The corresponding power frequency spectrum shows a comb structure, with each peak in the comb having its own comb, and so on, to finer and finer scales.