AI-powered edge devices currently lack the ability to adapt their embedded inference models to the ever-changing environment. To tackle this issue, Continual Learning (CL) strategies aim at incrementally improving the decision capabilities based on newly acquired data. In this work, after quantifying memory and computational requirements of CL algorithms, we define a novel HW/SW extreme-edge platform featuring a low power RISC-V octa-core cluster tailored for on-demand incremental learning over locally sensed data. The presented multi-core HW/SW architecture achieves a peak performance of 2.21 and 1.70 MAC/cycle, respectively, when running forward and backward steps of the gradient descent. We report the trade-off between memory footprint, latency, and accuracy for learning a new class with Latent Replay CL when targeting an image classification task on the CORe50 dataset. For a CL setting that retrains all the layers, taking 5h to learn a new class and achieving up to 77.3% of precision, a more efficient solution retrains only part of the network, reaching an accuracy of 72.5% with a memory requirement of 300 MB and a computation latency of 1.5 hours. On the other side, retraining only the last layer results in the fastest (867 ms) and less memory hungry (20 MB) solution but scoring 58% on the CORe50 dataset. Thanks to the parallelism of the low-power cluster engine, our HW/SW platform results 25x faster than typical MCU device, on which CL is still impractical, and demonstrates an 11x gain in terms of energy consumption with respect to mobile-class solutions.