We study the linear stability of plane Poiseuille flow of a Bingham fluid through a deformable neo-Hookean channel. The analysis reveals the existence of a finite-wave instability in the creeping-flow limit, which is absent for Newtonian fluid flow in the same geometry. Indeed, the instability in plane Poiseuille flow of a Bingham fluid closely resembles the finite-wave instability exhibited by plane Couette flow of a Newtonian fluid past a deformable solid. The motion of the unyielded region near the channel centerline effectively results in a Couette-like drag flow of the fluid in the yielded region, albeit with a parabolic base flow velocity profile. More importantly, the boundary conditions at the interface between yielded and unyielded zones is found to be crucial for the existence of the finite-wave instability. This is in stark contrast to Newtonian channel flow where the conditions at the channel centerline are dictated by the symmetry or antisymmetry of the velocity perturbations. The continuation of the predicted instability in the creeping-flow limit is also shown to determine the stability of the flow of a Bingham fluid at higher Reynolds number. In addition to treating the plug as a rigid solid, we also consider the plug to be an elastic solid and show that the elastic nature of the plug does not have any effect on the predicted finite-wave instability. We further argue, using the elastic plug scenario, that some of the regularized models can introduce model-dependent instabilities which are specific to the parameters used in the models. © 2019 American Physical Society.