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Continual learning, also known as lifelong learning, is an emerging research topic that has been attracting increasing interest in the field of machine learning. With human activity recognition (HAR) playing a key role in enabling numerous real-world applications, an essential step towards the long-term deployment of such recognition systems is to extend the activity model to dynamically adapt to changes in people's everyday behavior. Current research in continual learning applied to HAR domain is still under-explored with researchers exploring existing methods developed for computer vision in HAR. Moreover, analysis has so far focused on task-incremental or class-incremental learning paradigms where task boundaries are known. This impedes the applicability of such methods for real-world systems since data is presented in a randomly streaming fashion. To push this field forward, we build on recent advances in the area of continual machine learning and design a lifelong adaptive learning framework using Prototypical Networks, LAPNet-HAR, that processes sensor-based data streams in a task-free data-incremental fashion and mitigates catastrophic forgetting using experience replay and continual prototype adaptation. Online learning is further facilitated using contrastive loss to enforce inter-class separation. LAPNet-HAR is evaluated on 5 publicly available activity datasets in terms of the framework's ability to acquire new information while preserving previous knowledge. Our extensive empirical results demonstrate the effectiveness of LAPNet-HAR in task-free continual learning and uncover useful insights for future challenges.