In this manuscript, the (2+1)-dimensional Bogoyavlenskii's breaking soliton (BBS) model is considered. At-first, we reduce the model into its bilinear form using the Hirota bilinear approach. We then analytically construct lump waves and collision of lump with periodic waves via the Hirota scheme. We also present collision between lump wave and single-, double-kink soliton solutions, and the collision among lump, periodic and single-, double-kink soliton solutions of the model. In addition, we explain the fission properties of the collisions. It is noticed that collision of lump-kink waves split into double kinky-lump waves and gradually increases the number of such waves as the increase of λ, which was not found in the previous literature. Finally, we graphically present the nature of the collision solutions of the model in 3D and contour plots. The derived such wave solutions may have much more important for controlling unpredictable harmful waves arises in nature. •The (2+1)-dimensional Bogoyavlenskii's breaking soliton (BBS) model.•Bilinear form of the model by using the Hirota bilinear approach.•The collision among lump, periodic and kink soliton solutions are investigated.•Fission properties of the lump and periodic waves have been also observed.•Physical natures of the results have been analyzed and depicted graphically.