Hyperspectral videos can provide the spatial, spectral, and motion information of targets, which makes it possible to track camouflaged targets that are similar to the background. However, hyperspectral object tracking is a challenging task, due to the huge hyperspectral video data dimension and the "data hungry" problem for the model training. Insufficient training data can seriously interfere with the accuracy and generalization of the tracking models. In this paper, a dual deep Siamese network framework for hyperspectral object tracking (SiamHYPER) is proposed for learning a hyperspectral tracker from a pretrained RGB tracker in the case of the "data hungry" problem. Specifically, in addition to a pretrained RGB-based Siamese tracker, a hyperspectral target-aware module is designed to mine the spectral information during the target prediction, and a spatial-spectral cross-attention module is introduced to further fuse the deep spatial and spectral features extracted from the RGB tracker and the hyperspectral target-aware module. Benefiting from the guidance training of the RGB tracker, a robust hyperspectral object tracker can be trained effectively with only a small number of hyperspectral video samples, to overcome the "data hungry" problem. In the experiments conducted in this study, the SiamHYPER framework was verified using SiamBAN and SiamRPN++, with 13 000 frames of hyperspectral videos for training, and achieved the best performance on the publicly available hyperspectral dataset released as part of the WHISPERS Hyperspectral Object Tracking Challenge. The area under the curve (AUC) of SiamHYPER was increased by nearly 8.9% and 7.2%, respectively, when compared with the current state-of-the-art RGB-based and hyperspectral trackers. In addition, the processing speed of SiamHYPER was 19 FPS, which is much higher than that of the current state-of-the-art hyperspectral trackers. The source code is available at zhenliuzhenqi/HOT: Hyperspectral object tracking (github.com).