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Recent advances have shown that SNN-based systems can efficiently perform unsupervised continual learning due to their bio-plausible learning rule, e.g., Spike-Timing-Dependent Plasticity (STDP). Such learning capabilities are especially beneficial for use cases like autonomous agents (e.g., robots and UAVs) that need to continuously adapt to dynamically changing scenarios/environments, where new data gathered directly from the environment may have novel features that should be learned online. Current state-of-the-art works employ high-precision weights (i.e., 32 bit) for both training and inference phases, which pose high memory and energy costs thereby hindering efficient embedded implementations of such systems for battery-driven mobile autonomous systems. On the other hand, precision reduction may jeopardize the quality of unsupervised continual learning due to information loss. Towards this, we propose lpSpikeCon, a novel methodology to enable low-precision SNN processing for efficient unsupervised continual learning on resource-constrained autonomous agents/systems. Our lpSpikeCon methodology employs the following key steps: (1) analyzing the impacts of training the SNN model under unsupervised continual learning settings with reduced weight precision on the inference accuracy; (2) leveraging this study to identify SNN parameters that have a significant impact on the inference accuracy; and (3) developing an algorithm for searching the respective SNN parameter values that improve the quality of unsupervised continual learning. The experimental results show that our lpSpikeCon can reduce weight memory of the SNN model by 8x (i.e., by judiciously employing 4-bit weights) for performing online training with unsupervised continual learning and achieve no accuracy loss in the inference phase, as compared to the baseline model with 32-bit weights across different network sizes.