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Infrared small target super-resolution (SR) aims to recover reliable and detailed high-resolution image with high-contrast targets from its low-resolution counterparts. Since the infrared small target lacks color and fine structure information, it is significant to exploit the supplementary information among sequence images to enhance the target. In this paper, we propose the first infrared small target SR method named local motion and contrast prior driven deep network (MoCoPnet) to integrate the domain knowledge of infrared small target into deep network, which can mitigate the intrinsic feature scarcity of infrared small targets. Specifically, motivated by the local motion prior in the spatio-temporal dimension, we propose a local spatio-temporal attention module to perform implicit frame alignment and incorporate the local spatio-temporal information to enhance the local features (especially for small targets). Motivated by the local contrast prior in the spatial dimension, we propose a central difference residual group to incorporate the central difference convolution into the feature extraction backbone, which can achieve center-oriented gradient-aware feature extraction to further improve the target contrast. Extensive experiments have demonstrated that our method can recover accurate spatial dependency and improve the target contrast. Comparative results show that MoCoPnet can outperform the state-of-the-art video SR and single image SR methods in terms of both SR performance and target enhancement. Based on the SR results, we further investigate the influence of SR on infrared small target detection and the experimental results demonstrate that MoCoPnet promotes the detection performance. The code is available at https://github.com/XinyiYing/MoCoPnet.