Due to security imperatives, biometrics has attracted a lot of attention in recent decades. Biometric recognition refers to the identification of individuals based on their physiological and/or behavioral traits. Among the various physiological traits, the palmprint modality, which contains rich biometric features, has become one of the essential features that prove their effectiveness in improving the biometric recognition system accuracy. In addition to the palmprint texture features, infrared light can capture the vein-net of the palm, an independent biometric trait called palm-vein. Fortunately, these two biometric modalities can be easily obtained with a multispectral device and thus used together to enhance the biometric system. In this paper, we attempt to extract deep biometric features using a Convolutional Neural Network (CNN) to develop an effective deep-learning based multispectral palmprint recognition system. The tests results of extensive experiments conducted on a large and public palmprint multispectral database show that the proposed scheme effectively improves recognition results, mainly when fusing spectral bands of biometric modality.