This work describes a novel formulation for the simulation of incompressible Navier–Stokes problems involving nonconforming discretizations of membrane-like bodies. The new proposal relies on the use of a modified finite element space within the elements intersected by the embedded geometry, which is represented by a discontinuous (or element-by-element) level set function. This is combined with a Nitsche-based imposition of the general Navier-slip boundary condition, to be intended as a wall law model. Thanks to the use of an alternative finite element space, the formulation is capable of reproducing exactly discontinuities across the embedded interface, while preserving the structure of the graph of the discrete matrix. The performance, accuracy and convergence of the new proposal is compared with analytical solutions as well as with a body fitted reference technique. Moreover, the proposal is tested against another similar embedded approach. Finally, a realistic application showcasing the possibilities of the method is also presented.