Consider a branching random walk on the integer lattice where each particle jumps to a random neighbor at rate kappa, and splits into two descendants at rate lambda. We add to this model a degeneracy, where for every site all occupants are simultaneously killed at rate 1. For this model we give a characterization for the parameters kappa and lambda where (quenched or annealed) survival has positive probability. Using this characterization we then discuss how the survival probability behaves as a function of the parameters. Joint work with Nina Gantert.