Next-generation Cosmic Microwave Background observations will scrutinize how it is lensed by the matter in the Universe, determined to see a signature of neutrino mass. Here, we show that this lensing pattern also contains rich information on neutrino interactions: new neutrino interactions would modify the neutrino equation of state at different redshifts, shaping the late-time energy density in neutrinos to which lensing is sensitive. Intriguingly, the so-called lensing anomaly (the Planck satellite observes stronger CMB gravitational lensing than expected at ~3σ) could be explained by yet-undiscovered neutrino long-range interactions. Our results thus open up a window to link anomalous CMB lensing with present and future cosmological, astrophysical, and laboratory measurements of neutrino properties.