This paper investigates the bottom-hole pressure (BHP) performance of a fractured well with multiple radial fracture wings in a coalbed methane (CBM) reservoir with consideration of stress sensitivity. The fluid flow in the matrix simultaneously considers adsorption–desorption and diffusion, whereas fluid flow in the natural fracture system and the induced fracture network obeys Darcy’s law. The continuous line-source function in the CBM reservoir associated with the discretization method is employed in the Laplace domain. With the aid of Stehfest numerical inversion technology and Gauss elimination, the transient BHP responses are determined and analyzed. It is found that the main flow regimes for the proposed model in the CBM reservoir are as follows: linear flow between adjacent radial fracture wings, pseudo-radial flow in the inner region or Stimulated Reservoir Volume (SRV), and radial flow in outer region (un-stimulated region). The effects of permeability modulus, radius of SRV, ratio of permeability in SRV to that in un-stimulated region, properties of radial fracture wings, storativity ratio of the un-stimulated region, inter-porosity flow parameter, and adsorption–desorption constant on the transient BHP responses are discussed. The results obtained in this study will be of great significance for the quantitative analyzing of the transient performances of the wells with multiple radial fractures in CBM reservoirs.