Handwritten Signature Verification (HSV) systems distinguish between genuine and forged signatures. Traditional HSV development involves a static batch configuration, constraining the system's ability to model signatures to the limited data available. Signatures exhibit high intra-class variability and are sensitive to various factors, including time and external influences, imparting them a dynamic nature. This paper investigates the signature learning process within a data stream context. We propose a novel HSV approach with an adaptive system that receives an infinite sequence of signatures and is updated over time. Experiments were carried out on GPDS Synthetic, CEDAR, and MCYT datasets. Results demonstrate the superior performance of the proposed method compared to standard approaches that use a Support Vector Machine as a classifier. Implementation of the method is available at https://github.com/kdMoura/stream_hsv.