Accurately assessing the state of charge (SOC) of lithium-ion batteries is crucial for ensuring reliable and optimal functioning of electric vehicles (EVs). Essentially, an accurate SOC estimation allows the batteries to work in their desirable ranges. This paper introduces a new approach to SOC estimation by developing an $L_1$ robust observer for EV batteries. The proposed work's novelty and contribution are: First, unlike conventional $H_{\mathrm{\infty }}$ or generalized- $H_2$ robust methods that address energy-to-energy $(l_2-\text{to}-l_2)$ or energy-to-peak $(l_2$ -to- $l_{\mathrm{\infty }})$ disturbance attenuations, the $L_1$ robust algorithm aims to minimize peak-to-peak induced norm from the disturbances to the error indicator. Second, to the authors' best knowledge, this is the first-ever attempt to apply the $L_1$ robust observer to battery SOC estimation. By leveraging experimentally collected data, we validate the efficacy of our approach, which exhibits a better accuracy than a baseline algorithm. The devised solution has considerable potential to advance research and optimization in the field of battery management systems (BMS).