Recently, deep learning-based methods are proposed for hyperspectral images (HSIs) denoising. Among them, unsupervised methods such as deep image prior (DIP)-based methods have received much attention because these methods do not require any training data. However, DIP-based methods suffer from the semiconvergence behavior, i.e., the iteration of DIP-based methods needs to terminate by referring to the ground-truth image at the optimal iteration point. In this article, we propose the spatial-spectral constrained deep image prior (S2DIP) for the HSI mixed noise removal. Specifically, we integrate the DIP, the spatial-spectral total variation regularization term, and the $\ell _1$-norm sparse term to respectively capture the deep prior of the clean HSI, the spatial-spectral local smooth prior of the clean HSI, and the sparse prior of noise. The proposed S2DIP jointly leverages the expressive power brought from the deep convolutional neural network without any training data and exploits the HSI and noise structures via hand-crafted priors. Thus, our method avoids the semiconvergence behavior of DIP-based methods. Meanwhile, our method largely enhances the HSI denoising ability of DIP-based methods. To tackle the corresponding model, we utilize the alternating direction multiplier method algorithm. Extensive experiments demonstrate that our method outperforms model-based and deep learning-based state-of-the-art HSI denoising methods.