  We study a branching random walk with independent and identically
distributed, heavy tailed displacements. The offspring law is supercritical and
satisfies the Kesten-Stigum condition. We treat the case when the law of the
displacements does not lie in the max-domain of attraction of an extreme value
distribution. Hence, the classical extreme value theory, which is often
deployed in this kind of models, breaks down. We show that if the tails of the
displacements are such that the absolute value of the logarithm of the tail is
a slowly varying function, one can still effectively analyse the extremes of
the process. More precisely, after a non-linear transformation the extremes of
the branching random walk process converge to a cluster Cox process.
