Abstract In this paper, we propose and analyze a $C^1$-virtual element method of high order to solve the Brinkman problem formulated in terms of the stream function. The velocity is obtained as a simple post-process from stream function and a novel strategy is written to recover the fluid pressure. We establish optimal a priori error estimates for the stream function, velocity and pressure with constants independent of the viscosity. Finally, we report some numerical test illustrating the behavior of the virtual scheme and supporting our theoretical results on different families of polygonal meshes.