Consider a critical branching Lévy process $\{X_t, t\ge 0\}$ with branching rate $β>0, $ offspring distribution $\{p_k:k\geq 0\}$ and spatial motion $\{ξ_t, Π_x\}$. For any $t\ge 0$, let $N_t$ be the collection of particles alive at time $t$, and, for any $u\in N_t$, let $X_u(t)$ be the position of $u$ at time $t$. We study the tail probability of the maximal displacement $M:=\sup_{t>0}\sup_{u\in N_t} X_u(t)$ under the assumption $\lim_{n\to\infty} n^α\sum_{k=n}^\infty p_k =κ\in(0,\infty)$ for some $α\in (1,2)$, $Π_0(ξ_1)=0$ and $Π_0 (|ξ_1|^r)\in (0,\infty)$ for some $r> 2α/(α-1)$. Our main result is a generalization of the main result of Sawyer and Fleischman (1979) for branching Brownian motions and that of Lalley and Shao (2015) for branching random walks, both of which are proved under the assumption $\sum_{k=0}^\infty k^3 p_k<\infty$.