Terrestrial planets in the solar system, such as the Earth, are oxygen-rich, with silicates and iron being the most common minerals in their interiors. However, the true chemical diversity of rocky planets orbiting other stars is yet unknown. Mass and radius measurements are used to constrain the interior compositions of super-Earths (exoplanets with masses of 1 - 10 Earth masses), and are typically interpreted with planetary interior models that assume Earth-centric oxygen-rich compositions. Using such models, the super-Earth 55 Cancri e (mass of 8 Earth masses, radius of 2 Earth radii) has been suggested to bear an interior composition consisting of Fe, silicates, and an envelope (>= 10% by mass) of super-critical water. We report that the mass and radius of 55 Cancri e can also be explained by a carbon-rich solid interior made of Fe, C, SiC, and/or silicates and without a volatile envelope. While the data allow Fe mass fractions of up to 40%, a wide range of C, SiC and/or silicate mass fractions are possible. A carbon-rich 55 Cancri e is also plausible if its protoplanetary disk bore the same composition as its host star, which has been reported to be carbon-rich. However, more precise estimates of the stellar elemental abundances and observations of the planetary atmosphere are required to further constrain its interior composition. The possibility of a C-rich interior in 55 Cancri e opens a new regime of geochemistry and geophysics in extraterrestrial rocky planets, compared to terrestrial planets in the solar system.