A nonlinear integrable model known as the (4 + 1)-dimensional Boiti–Leon–Manna–Pempinelli (4D-BLMP) equation is studied in the present paper. To this end, by considering the Hirota bilinear form of the model and utilizing the linear superposition method (LSM) along with symbolic computations, a group of rational wave solutions including multiple wave and positive (non-singular) compelexiton solutions is formally derived. The dynamical behavior of the solutions is also analyzed graphically by considering the special values of the involved parameters. The results of the current work reveal the existence of different wave structures to the 4D-BLMP equation and distinguish it from other models that do not possess non-singular compelexiton solutions.