  In this paper, we study the weak convergence of the extremes of supercritical
branching L\'evy processes $\{\mathbb{X}_t, t \ge0\}$ whose spatial motions are
L\'evy processes with regularly varying tails. The result is drastically
different from the case of branching Brownian motions. We prove that, when
properly renormalized, $\mathbb{X}_t$ converges weakly. As a consequence, we
obtain a limit theorem for the order statistics of $\mathbb{X}_t$.
