It has been proved that surface plasmon polariton (SPP) can well conserve and transmit the quantum nature of entangled photons. Therefore, further utilization and manipulation of such quantum nature of SPP in a plasmonic chip will be the next task for scientists in this field. In quantum logic circuits, the controlled-NOT (CNOT) gate is the key building block. Here, we implement the first plasmonic quantum CNOT gate with several-micrometer footprint by utilizing a single polarization-dependent beam-splitter (PDBS) fabricated on the dielectric-loaded SPP waveguide (DLSPPW). The quantum logic function of the CNOT gate is characterized by the truth table with an average fidelity of. Its entangling ability to transform a separable state into an entangled state is demonstrated with the visibilities of and for non-orthogonal bases. The DLSPPW based CNOT gate is considered to have good integratability and scalability, which will pave a new way for quantum information science.