SuperSUN, a new superthermal source of ultracold neutrons (UCN) at the Institut Laue-Langevin, exploits inelastic scattering of neutrons in isotopically pure superfluid ${}^4$He at temperatures below $0.6$K. For the first time, continuous operation with an intense broad-spectrum cold neutron beam is demonstrated over 60 days. We observe continuous UCN extraction rates of $21000$s${}^{-1}$, and storage in the source with saturated density $273$cm${}^{-3}$. The low-energy UCN spectrum is alterable via accumulation and holding delays, opening new possibilities in fundamental and applied physics.