T-carbon had been proposed as a new carbon allotrope in 2011, which was successfully synthesized in recent experiments. Because of its flfluffy structure, several kinds of atoms can be intercalated into T-carbon, making it a potential versatile candidate in various applications such as hydrogen storage, solar cells, lithium ion batteries, thermoelectrics, photocatalyst, etc. Here we show that superconductivity can appear in Na-doped T-carbon with a superconducting transition temperature Tc of 11 K at ambient pressure, and Tc can be enhanced to 19 K under a pressure of 14 GPa. The calculations on temperature dependence of specifific heat and electrical and thermal conductivities show that the normal state of the Na-doped T-carbon superconductor is a non-Fermi liquid at temperature below 50 K, where the Wiedemann-Franz law is remarkably violated. The prediction of superconductivity in Na-doped T-carbon would spur great interest both experimentally and theoretically to explore novel carbon-based superconductors.