This work addresses the problem of anonymizing the identity of faces in a dataset of images, such that the privacy of those depicted is not violated, while at the same time the dataset is useful for downstream task such as for training machine learning models. To the best of our knowledge, we are the first to explicitly address this issue and deal with two major drawbacks of the existing state-of-the-art approaches, namely that they (i) require the costly training of additional, purpose-trained neural networks, and/or (ii) fail to retain the facial attributes of the original images in the anonymized counterparts, the preservation of which is of paramount importance for their use in downstream tasks. We accordingly present a task-agnostic anonymization procedure that directly optimises the images’ latent representation in the latent space of a pre-trained GAN. By optimizing the latent codes directly, we ensure both that the identity is of a desired distance away from the original (with an identity obfuscation loss), whilst preserving the facial attributes (using a novel feature-matching loss in FaRL’s deep feature space). We demonstrate through a series of both qualitative and quantitative experiments that our method is capable of anonymizing the identity of the images whilst--crucially--better-preserving the facial attributes. We make the code and the pre-trained models publicly available at: https://github.com/chi0tzp/FALCO.