  Consider a critical branching random walk on $\mathbb{Z}^d$, $d\geq 1$,
started with a single particle at the origin, and let $L(x)$ be the total
number of particles that ever visit a vertex $x$. We study the tail of $L(x)$
under suitable conditions on the offspring distribution. In particular, our
results hold if the offspring distribution has an exponential moment.
