  We demonstrate that interactions can substantially undermine the
free-particle description of magnons in ferromagnets on geometrically
frustrated lattices. The anharmonic coupling, facilitated by the
Dzyaloshinskii-Moriya interaction, and a highly-degenerate two-magnon continuum
yield a strong, non-perturbative damping of the high-energy magnon modes. We
provide a detailed account of the effect for the $S=1/2$ ferromagnet on the
kagom\'e lattice and propose further experiments.
