Interfaces between materials with differently ordered phases present unique opportunities for exotic physical properties, especially the interplay between ferromagnetism and superconductivity in the ferromagnet/superconductor heterostructures. The investigation of zero- and π-junctions has been of particular interest for both fundamental physical science and emerging technologies. Here, we report the experimental observation of giant oscillatory Gilbert damping in the superconducting niobium/nickel-iron/niobium junctions with respect to the nickel-iron thickness. This observation suggests an unconventional spin pumping and relaxation via zero-energy Andreev bound states that exist not only in the niobium/nickel-iron/niobium π-junctions but also in the niobium/nickel-iron/niobium zero-junctions. Our findings could be important for further exploring the exotic physical properties of ferromagnet/superconductor heterostructures and potential applications of ferromagnet π-junctions in quantum computing, such as half-quantum flux qubits.