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 $ζ^{(0,\infty)}$ 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 $M^{(0,\infty)}: = \sup_{t\geq 0} M_t^{(0,\infty)}$. Under the assumption that the offspring distribution belongs to the domain of attraction of an $α$-stable distribution, $α\in (1, 2]$, and some moment conditions on the spatial motion, we give the decay rates of the survival probabilities $$ \mathbb{P}_{y}(ζ^{(0,\infty)}>t), \quad \mathbb{P}_{\sqrt{t}y}(ζ^{(0,\infty)}>t) $$ and the tail probabilities $$ \mathbb{P}_{y}(M^{(0,\infty)}\geq x), \quad \mathbb{P}_{xy}(M^{(0,\infty)}\geq x). $$ We also study the scaling limits of $M_t^{(0,\infty)}$ and the point process $Z_t^{(0,\infty)}$ under $\mathbb{P}_{\sqrt{t}y}(\cdot |ζ^{(0,\infty)}>t)$ and $\mathbb{P}_y(\cdot |ζ^{(0,\infty)}>t)$. The scaling limits under $\mathbb{P}_{\sqrt{t}y}(\cdot |ζ^{(0,\infty)}>t)$ are represented in terms of super killed Brownian motion.