In this paper, we study asymptotic behaviors of the tails of extinction time and maximal displacement of a critical branching killed Lévy process \((Z_t^{(0,\infty )})_{t\ge 0}\) in \(\mathbb {R} \) , in which all particles (and their descendants) are killed upon exiting \((0, \infty )\) . Let \(\zeta ^{(0,\infty )}\) and \(M_t^{(0,\infty )}\) be the extinction time and maximal position of all the particles alive at time t of this branching killed Lévy process and define \(M^{(0,\infty )}: = \sup _{t\ge 0} M_t^{(0,\infty )}\) . Under the assumption that the offspring distribution belongs to the domain of attraction of an \(\alpha \) -stable distribution, \(\alpha \in (1, 2]\) , and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities \( \mathbb {P} _y(\zeta ^{(0,\infty )}>t), \quad \mathbb {P} _{\sqrt{t}y}(\zeta ^{(0,\infty )}>t) \) and the tail probabilities \( \mathbb {P} _{y}(M^{(0,\infty )}\ge x), \quad \mathbb {P} _{xy}(M^{(0,\infty )}\ge x). \) We also study the scaling limits of \(M_t^{(0,\infty )}\) and the point process \(Z_t^{(0,\infty )}\) under \(\mathbb {P} _{\sqrt{t}y}(\cdot |\zeta ^{(0,\infty )}>t)\) and \(\mathbb {P} _y(\cdot |\zeta ^{(0,\infty )}>t)\) . The scaling limits under \(\mathbb {P} _{\sqrt{t}y}(\cdot |\zeta ^{(0,\infty )}>t)\) are represented in terms of super killed Brownian motion.
