We propose Black-Box IoT (BBox-IoT), a new ultra-lightweight black-box system for authenticating and storing IoT data. BBox-IoT is tailored for deployment on IoT devices (including low-Size Weight and Power sensors) which are extremely constrained in terms of computation, storage, and power. By utilizing core Blockchain principles, we ensure that the collected data is immutable and tamperproof while preserving data provenance and non-repudiation. To realize BBox-IoT, we designed and implemented a novel chain-based hash signature scheme which only requires hashing operations and removes all synchronicity dependencies between signer and verifier. Our approach enables low-SWaP devices to authenticate removing reliance on clock synchronization. Our evaluation results show that BBox-IoT is practical in Industrial Internet of Things (IIoT) environments: even devices equipped with 16MHz microcontrollers and 2KB memory can broadcast their collected data without requiring heavy cryptographic operations or synchronicity assumptions. Finally, when compared to industry standard ECDSA, our approach is two and three orders of magnitude faster for signing and verification operations respectively. Thus, we are able to increase the total number of signing operations by more than 5000% for the same amount of power.