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In the past few years, there has been a leap from traditional palmprint recognition methodologies, which use handcrafted features, to deep-learning approaches that are able to automatically learn feature representations from the input data. However, the information that is extracted from such deep-learning models typically corresponds to the global image appearance, where only the most discriminative cues from the input image are considered. This characteristic is especially problematic when data is acquired in unconstrained settings, as in the case of contactless palmprint recognition systems, where visual artifacts caused by elastic deformations of the palmar surface are typically present in spatially local parts of the captured images. In this study we address the problem of elastic deformations by introducing a new approach to contactless palmprint recognition based on a novel CNN model, designed as a two-path architecture, where one path processes the input in a holistic manner, while the second path extracts local information from smaller image patches sampled from the input image. As elastic deformations can be assumed to most significantly affect the global appearance, while having a lesser impact on spatially local image areas, the local processing path addresses the issues related to elastic deformations thereby supplementing the information from the global processing path. The model is trained with a learning objective that combines the Additive Angular Margin (ArcFace) Loss and the well-known center loss. By using the proposed model design, the discriminative power of the learned image representation is significantly enhanced compared to standard holistic models, which, as we show in the experimental section, leads to state-of-the-art performance for contactless palmprint recognition. Our approach is tested on two publicly available contactless palmprint datasets—namely, IITD and CASIA—and is demonstrated to perform favorably against state-of-the-art methods from the literature. The source code for the proposed model is made publicly available.