We show that electron-magnon interactions at a ferromagnetic metal-superconductor interface lead to a new process of magnon-assisted Andreev reflection, which consists of the simultaneous injection of a Cooper pair from the superconductor and the emission of a magnon inside the ferromagnet. At low temperature this process represents an additional channel for subgap transport across an FS interface, which lifts restrictions on the current I resulting from the necessity to match spin-polarised current in the ferromagnet with spin-less current in the superconductor. We calculate I using the tunnelling Hamiltonian method and the nonequilibrium (Keldysh) Green functions technique. It is shown that the inelastic magnon-assisted Andreev process would manifest itself as a nonlinear addition to the I(V) characteristics which is asymmetric with respect to the sign of the bias voltage and is related to the local density of states of magnons at the interface.