  In the ballistic annihilation process, particles on the real line have
independent speeds symmetrically distributed in $\{-1,0,+1\}$ and are
annihilated by collisions. It is widely believed that there is a phase
transition at $p=p_{\mathrm c}=0.25$ between regimes where every particle is
eventually annihilated and where some particles survive forever, where $p$ is
the proportion of stationary particles. It is easy to see that some particles
survive if $p>0.5$, and rigorous proofs giving better upper bounds on
$p_{\mathrm c}$ have recently appeared. However, no nontrivial lower bound on
$p_{\mathrm c}$ was previously known. We prove that $p_{\mathrm c}\geq
0.21699$, and give a comparable bound for a discretised version.
