We propose a process of magnon-assisted Andreev reflection at a ferromagnetic metal-superconductor (FS) interface, 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 resulting from the necessity to match spin-polarized current in the ferromagnet with spinless current in the superconductor. For a junction between a superconductor and a ferromagnet with an arbitrary degree of polarization, the inelastic magnon-assisted Andreev reflection 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 density of states of magnons in the ferromagnet. Expressions for the subgap I(V) characteristics are given for arbitrary interfacial quality while the limiting cases of uniformly transparent and disordered interfaces are discussed in detail.