A (2+1)-dimensional extended Calogero–Bogoyavlenskii–Schiff system in nonlinear optics, fluid dynamics and plasma physics is investigated via the symbolic computation in this paper. Soliton solutions, which are kink-shaped, are obtained via the Hirota method. Breather solutions are derived via the extended homoclinic test approach, and lump solutions are obtained from the breather solutions under a limiting procedure. We find that the shape and amplitude of the one soliton keep unchanged during the propagation, and the velocity of one soliton depends on all the coefficients in the system. We graphically demonstrate that the interaction between the two solitons is elastic, and analyse the solitons with the influence of the coefficients. We observe that the amplitudes and shapes of the breather and lump remain unchanged during the propagation, and graphically present the breathers and lumps with the influence of the coefficients.