We introduce the notion of Zimmer amenability for actions of discrete quantum groups on von Neumann algebras. We prove non‑commuta-tive generalizations of several fundamental results from the classical setting. In particular, we characterize Zimmer amenability of such an action in terms of injectivity of the associated von Neumann algebra crossed product as a module over the dual quantum group. As an application, we show that actions of any discrete quantum group on its Poisson boundaries are always Zimmer amenable.

We are grateful to Massoud Amini for his continuous encouragement throughout this project. We would also like to thank Mehrdad Kalantar and Jason Crann for their helpful comments.