The Witten deformation on a smooth compact manifold is an analytic proof of the Morse inequalities, which has been proposed by Witten in the 80s and is inspired from ideas in quantum field theory. The Witten deformation is one of the main actors in the extension by Bismut and Zhang of the comparison between analytic and topological torsion of a smooth compact manifold, aka the Cheeger-Müller theorem. The aim of this talk is to explain how to extend the Witten deformation to singular spaces with conical singularities and radial Morse functions, and how this can be used to achieve a Cheeger-Müller theorem for these spaces.