Freezing of dispersions is omnipresent in science and technology. While the passing of a freezing front over a solid particle is reasonably understood, this is not so for soft particles. Here, using an oil-in-water emulsion as a model system, we show that when engulfed into a growing ice front, a soft particle severely deforms. This deformation strongly depends on the engulfment velocity V, even forming pointy-tip shapes for low values of V. We find such singular deformations are mediated by interfacial flows in nanometric thin liquid films separating the nonsolidifying dispersed droplets and the solidifying bulk. We model the fluid flow in these intervening thin films using a lubrication approximation and then relate it to the deformation sustained by the dispersed droplet.