We present spectroscopy of the ultra-faint Milky Way satellites Eridanus III (Eri III) and DELVE 1. We identify eight member stars in each satellite and place nonconstraining upper limits on their velocity and metallicity dispersions. The brightest star in each object is very metal poor, at [Fe/H] = −3.1 for Eri III and [Fe/H] = −2.8 for DELVE 1. Both of these stars exhibit large overabundances of carbon and very low abundances of the neutron-capture elements Ba and Sr, and we classify them as CEMP-no stars. Because their metallicities are well below those of the Milky Way globular cluster population, and because no CEMP-no stars have been identified in globular clusters, these chemical abundances could suggest that Eri III and DELVE 1 are dwarf galaxies. On the other hand, the two systems have half-light radii of 8 pc and 6 pc, respectively, which are more compact than any known ultra-faint dwarfs. We conclude that Eri III and DELVE 1 are either the smallest dwarf galaxies yet discovered, or they are representatives of a new class of star clusters that underwent chemical evolution distinct from that of ordinary globular clusters. In the latter scenario, such objects are likely the most primordial star clusters surviving today. These possibilities can be distinguished by future measurements of carbon and/or iron abundances for larger samples of stars or improved stellar kinematics for the two systems.