Currently, there is much interest in the study on the localized waves and their interactions in fluid mechanics. An extended (3+1)-dimensional Kadomtsev-Petviashvili equation in fluid mechanics is considered here. By means of the Hirota method, we obtain the N-soliton solutions, with N as a positive integer. The higher-order breather solutions are obtained from the N-soliton solutions employing the complex conjugated transformations. Making use of the long-wave limit method, we determine the higher-order lump solutions from the N-soliton solutions. Besides, some hybrid solutions are presented. Three kind of the localized waves, namely, the solitons, breathers and lumps, along with their interactions, are investigated via the above solutions. Amplitudes, shapes and velocities of those localized waves remain invariant after the interactions, which indicates the interactions are elastic.