Unsupervised fusion-based hyperspectral imagery (HSI) super-resolution (SR) is an essential task of HSI processing, which aims to reconstruct a high-resolution (HR) HSI using only an observed low-resolution HSI and a conventional HR image. Although a large number of unsupervised HSI SR methods have been proposed, the heuristic handcrafted image priors adopted by the majority of these methods restrict their capacity to capture specific characteristics of the HSI, as well as their ability to generalize to noisy observation images. In this study, we investigate a fusion-based HSI SR framework with the deep image prior, in which the deep neural network (rather than a heuristic handcrafted image prior) is exploited to capture plenty of image statistics. Within this framework, we further propose an unsupervised recurrence-based HSI SR method using pixel-aware refinement, which utilizes the intermediate reconstruction results to self-supervise unsupervised learning. Due to containing the information of the image-specific characteristic, the proposed method achieves better performance, in terms of both accuracy and robustness to noise, compared with the existing methods. Extensive experiments on four HSI data sets demonstrate the effectiveness of the proposed method.