In functional data analysis, binary classification with one functional covariate has been extensively studied. We aim to fill in the gap of considering multivariate functional covariates in classification. In particular, we propose an $L_1$-regularized functional support vector machine for binary classification. An accompanying algorithm is developed to fit the classifier. By imposing an $L_1$ penalty, the algorithm enables us to identify relevant functional covariates of the binary response. Numerical results from simulations and one real-world application demonstrate that the proposed classifier enjoys good performance in both prediction and feature selection.