The topological properties of massive and massless fermionic quasiparticles have been intensively investigated over the past decade in topological materials without magnetism. Recently, the bosonic analogs of such quasiparticles arising from spin waves have been reported in a two-dimensional (2D) honeycomb-lattice ferromagnet/antiferromagnet and a 3D antiferromagnet. Here, we use time-of-flight inelastic neutron scattering to study spin waves of the S=1 honeycomb-lattice antiferromagnet BaNi2(AsO4)2, which has a zigzag antiferromagnetic (AFM) ground state identical to that of the Kitaev quantum spin liquid candidate α−RuCl3. We determine the magnetic exchange interactions in the zigzag AFM ordered phase, and show that spin waves in BaNi2(AsO4)2 have symmetry-protected Dirac points inside the Brillouin zone boundary. These results provide a microscopic understanding of the zigzag AFM order and associated Dirac magnons in honeycomb-lattice magnets, and are also important for establishing the magnetic interactions in Kitaev quantum spin liquid candidates.