Let $(W_n(θ))_{n \in \mathbb{N}_0}$ be Biggins' martingale associated with a supercritical branching random walk, and let $W(θ)$ be its almost sure limit. Under a natural condition for the offspring point process in the branching random walk, we show that if the law of $W_1(θ)$ belongs to the domain of normal attraction of an $α$-stable distribution for some $α\in (1,2)$, then, as $n\to\infty$, there is weak convergence of the tail process $(W(θ) - W_{n-k}(θ))_{k \in \mathbb{N}_0}$, properly normalized, to a random scale multiple of a stationary autoregressive process of order one with $α$-stable marginals.