The authors develop an arbitrary order mixed virtual element formulation (mixed VEM) for a coupled fractured porous medium. The fracture set consists of a set of 2D planar, possibly intersecting, polygons embedded in a 3D matrix representing the porous medium. The model equation consists of an elliptic equation for pressure in the porous matrix coupled to an elliptic equation in the fracture network leading to a multi@-dimensional model, called a hybrid-dimensional or mixed-dimensional model. This work extends previous works on mixed VEMs to the present case of a multi-dimensional model with an arbitrary order mixed VEM method. The use of a VEM simplifies the meshing procedure, and the mixed method ensures the local mass conservation. The article details the implementation of the method and discusses the accuracy of the solution for several interesting cases.