We study cut finite element discretizations of a Darcy interface problem based on the mixed finite element pairs RTk \times Qk, k \geq 0. Here Qk is the space of discontinuous polynomial functions of degree less than or equal to k and RT is the Raviart-Thomas space. We show that the standard ghost penalty stabilization, often added in the weak forms of cut finite element methods for stability and control of the condition number of the resulting linear system matrix, destroys the divergence-free property of the considered element pairs. Therefore, we propose new stabilization terms for the pressure and show that we recover the optimal approximation of the divergence without losing control of the condition number of the linear system matrix. We prove that with the new stabilization term the proposed cut finite element discretization results in pointwise divergence-free approximations of solenoidal velocity fields. We derive a priori error estimates for the proposed unfitted finite element discretization based on RTk \times Qk, k \geq 0. In addition, by decomposing the computational mesh into macroelements and applying ghost penalty terms only on interior edges of macroelements, stabilization is applied very restrictively and active only where needed. Numerical experiments with element pairs RT0 \times Q0, RT1 \times Q1, and BDM1 \times Q0 (where BDM is the Brezzi-Douglas-Marini space) indicate that with the new method we have (1) optimal rates of convergence of the approximate velocity and pressure; (2) well-posed linear systems where the condition number of the system matrix scales as it does for fitted finite element discretizations; (3) optimal rates of convergence of the approximate divergence with pointwise divergence-free approximations of solenoidal velocity fields. All three properties hold independently of how the interface is positioned relative to the computational mesh.