The brain-inspired Spiking Neural Networks (SNNs) work in an event-driven manner and have an implicit recurrence in neuronal membrane potential to memorize information over time, which are inherently suitable to handle temporal event-based streams. Despite their temporal nature and recent approaches advancements, these methods have predominantly been assessed on event-based classification tasks. In this paper, we explore the utility of SNNs for event-based tracking tasks. Specifically, we propose a brain-inspired adaptive Leaky Integrate-and-Fire neuron (BA-LIF) that can adaptively adjust the membrane time constant according to the inputs, thereby accelerating the leakage of meaningless noise features and reducing the decay of valuable information. SNNs composed of our proposed BA-LIF neurons can achieve high performance without a careful and time-consuming trial-by-error initialization on the membrane time constant. The adaptive capability of our network is further improved by introducing an extra temporal feature aggregator (TFA) that assigns attention weights over the temporal dimension. Extensive experiments on various event-based tracking datasets validate the effectiveness of our proposed method. We further validate the generalization capability of our method by applying it to other event-classification tasks.