  We consider branching random walks with a spine in the domain of attraction
of an $\alpha$-stable L\'evy process. For this process, the classical
derivative martingale in general degenerates in the limit. We first determine
the quantity replacing the derivative martingale and show that it converges to
a non-degenerate limit under a certain LlogL-type condition which we assume to
be optimal. We go on to give the Seneta-Heyde norming for the critical additive
martingale under the same assumptions. The proofs are based on the methods
introduced in our previous paper which considered the finite variance case
[Boutaud and Maillard (2019), EJP, vol. 24, paper no. 99].
