Cell-free massive MIMO (CF-mMIMO) networks have recently emerged as a promising solution to tackle the challenges arising from next-generation massive machine-type communications. In this paper, a fully grant-free deep learning (DL)-based method for user activity detection in CF-mMIMO networks is proposed. Initially, the known non-orthogonal pilot sequences are used to estimate the channel coefficients between each user and the access points. Then, a deep convolutional neural network is used to estimate the activity status of the users. The proposed method is "blind", i.e., it is fully data-driven and does not require prior large-scale fading coefficients estimation. Numerical results show how the proposed DL-based algorithm is able to merge the information gathered by the distributed antennas to estimate the user activity status, yet outperforming a state-of-the-art covariance-based method.