Source device identification is an important topic in image forensics since it allows to trace back the origin of an image. Its forensics counter-part is source device anonymization, that is, to mask any trace on the image that can be useful for identifying the source device. A typical trace exploited for source device identification is the Photo Response Non-Uniformity (PRNU), a noise pattern left by the device on the acquired images. In this paper, we devise a methodology for suppressing such a trace from natural images without significant impact on image quality. Specifically, we turn PRNU anonymization into an optimization problem in a Deep Image Prior (DIP) framework. In a nutshell, a Convolutional Neural Network (CNN) acts as generator and returns an image that is anonymized with respect to the source PRNU, still maintaining high visual quality. With respect to widely-adopted deep learning paradigms, our proposed CNN is not trained on a set of input-target pairs of images. Instead, it is optimized to reconstruct the PRNUfree image from the original image under analysis itself. This makes the approach particularly suitable in scenarios where large heterogeneous databases are analyzed and prevents any problem due to lack of generalization. Through numerical examples on publicly available datasets, we prove our methodology to be effective compared to state-of-the-art techniques.