Given a sound field generated by a sparse distribution of impulse image sources, can the continuous 3D positions and amplitudes of these sources be recovered from discrete, band-limited measurements of the field at a finite set of locations, e.g. , a multichannel room impulse response? Borrowing from recent advances in super-resolution imaging, it is shown that this non-linear, non-convex inverse problem can be efficiently relaxed into a convex linear inverse problem over the space of Radon measures in R^3 . The new linear operator introduced here stems from the fundamental solution of the wave equation combined with the receivers' responses. An adaptation of the Sliding Frank-Wolfe algorithm is proposed to numerically solve the problem off-the-grid , i.e. , in continuous 3D space. Idealized simulated experiments show that the approach can recover hundreds of image sources at a rate and accuracy that are not achievable by previous methods, using a compact microphone array and source placed at random in random-sized shoe-box rooms. The impact of noise, sampling rate and array diameter on these results is also examined.