Recent large-scale Byzantine-Fault-Tolerant (BFT) algorithms provide scalability at a low cost by exploiting a secure Random Peer Sampling (RPS) service: a service that provides a stream of random network nodes where no attacking entity can become over-represented. Unfortunately, producing good peer samples untainted by Byzantine behavior in a large-scale network is particularly difficult, with existing solutions unable to withstand aggressive attacks. In this paper, we propose a novel RPS algorithm, BASALT, that implements what we have termed a stubborn chaotic search over node IDs to counter attackers' attempts at becoming over-represented. Our evaluation based on a theoretical analysis, Monte Carlo simulations, and experiments on a live cryptocurrency network shows that BASALT delivers close-to-optimal protection against malicious behaviors and outperforms state-of-the-art solutions by a wide margin. CCS Concepts: • Software and its engineering → Distributed systems organizing principles; Peer-to-peer architectures; • Theory of computation → Distributed algorithms; • Computer systems organization → Dependable and fault-tolerant systems and networks.