We demonstrate that nonlinear response functions in many-body systems carry a sharp signature of interactions between gapped low-energy quasiparticles. Such interactions are challenging to deduce from linear response measurements. The signature takes the form of a divergent-in-time contribution to the response-linear in time in the case when quasiparticles propagate ballistically-that is absent for free bosonic excitations. We give a physically transparent semiclassical picture of this singular behavior. While the semiclassical picture applies to a broad class of systems we benchmark it in two simple models: in the Ising chain using a form-factor expansion, and in a nonintegrable model-the spin-1 Affleck-Kennedy-Lieb-Tasaki chain-using time-dependent density matrix renormalization group simulations. We comment on extensions of these results to finite temperatures.