A simple model of radiating diffusion flames considered by Kavousanakis et al. (2013) [1] is extended to two spatial dimensions. A large variety of stationary spatially localized states representing the breakup of the flame front near extinction is computed using numerical continuation. These states are organized by a global bifurcation in space that takes place at a particular value of the Damköhler number and their existence is consistent with current understanding of spatial localization in driven dissipative systems.