Deep learning-based methods, particularly Mamba-based models, hold vast potential for hyperspectral image super-resolution (HSI-SR) reconstruction. However, the intrinsic spatial-spectral details in hyperspectral images are hard to be well represented solely utilizing the hierarchical learning manner in most Mamba structures, which results in the degradation of restoration quality on inherent fine-grain textures. To address this challenge, this paper proposes a Spatial-Spectral Prior Guided Mamba Network (SSP-Mamba) for HSI-SR, which leverages spatial-spectral priors in Wavelet domain to refine detail recovery with cross-layer embedment. Specifically, this model extracts multi-scale spatial dependences with a Wavelet transform and yields spectral responses with a spectral attention mechanism, which adaptively exploits specific bands to acquires an instructive spatial-spectral joint prior. The prior is then injected into hierarchical deep features from the backbone network—built upon a Shuffle-ReShuffle Mamba (SRM) module—through a spatial selection mechanism, thereby enabling the impressive feature representation capability. Extensive experiments on three representative datasets show that SSP-Mamba model achieves superior HSI-SR performance to other deep learning approaches in terms of the quantitative and qualitive metrics.