We consider a system of particles in $\mathbb{R}^d$ performing symmetric stable motion with exponent $\alpha, 0 < \alpha \leq 2$, and branching at the end of an exponential lifetime with offspring generating function $F(s) = s + \frac{1}{2}(1 - s)^{1+\beta}, 0 < \beta \leq 1$. (This includes binary branching Brownian motion for $\alpha = 2, \beta = 1$.) It is shown that, for an initial Poisson population with uniform intensity, the system goes to extinction if $d \leq \alpha/\beta$ and is "persistent" (i.e., preserves intensity in the large time limit) if $d > \alpha/\beta$. To this purpose a continuous-time version of Kallenberg's backward technique for computing Palm distributions of branching particle systems is developed, which permits us to adapt methods used by Dawson and Fleischmann in the study of discrete-space and discrete-time systems.