This paper investigates rational solutions of an extended Camassa-Holm-Kadomtsev-Petviashvili equation, which simulates dispersion's role in the development of patterns in a liquid drop, and describes left and right traveling waves like the Boussinesq equation. Through its bilinear form and symbolic computation, we derive some multiple order rational and generalized rational solutions and analyze their dynamic features, such as the connection between rational solution and bilinear equation, scatter behavior, moving path, and exact location of the soliton. The obtained solutions demonstrate two wave forms: multi-lump and multi-wave that consist of three, six and eight lump waves or two, three and four line waves. Moreover, different from the multi-wave solitons, stationary multiple dark waves are presented.