As one type of local invariant feature, corner feature plays an important role in diverse applications such as: video mining, target detection, image classification, image retrieval, and image matching, etc. However, there are few studies on corner feature for hyperspectral image (HSI). Therefore, this paper proposes a novel corner feature for HSI named extreme-constrained spatial-spectral corner (ECSSC for short) and its corresponding detector. The definition of ECSSC is developed based on the definition of spectral-spatial interest point and the characteristic of HSI. Based on this definition, the detector of ECSSC is put forward and introduced in detail. Then, as an important application of ECSSC, an efficient framework for image-level HSI classification is designed based on ECSSC and parallel computation. The experimental results show that the proposed algorithm can detect abundant corner features with high repeatability rate from HSI and the accuracy of image-level HSI based on ECSSC is dramatically higher than that of the state of the art.