In this paper, a new (3+1)-dimensional integrable Kadomtsev–Petviashvili equation is developed. Its integrability is verified by the Painlev\’e analysis. The bilinear form, multiple-soliton, breather and lump solutions are obtained via using the Hirota bilinear method. Furthermore, the abundant dynamical behaviors for these solutions are discovered. It is interesting that there are splitting and fusing phenomena when the lumps interact.