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Context: A challenge with radial-velocity (RV) data is disentangling the origin of signals either due to a planetary companion or to stellar activity. In fact, the existence of a planetary companion has been proposed, as well as contested, around the relatively bright, nearby M3.0V star AD Leo at the same period as the stellar rotation of 2.23d. Aims: We further investigate the nature of this signal. We introduce new CARMENES optical and near-IR RV data and an analysis in combination with archival data taken by HIRES and HARPS, along with more recent data from HARPS-N, GIANO-B, and HPF. Also, we address the confusion concerning the binarity of AD Leo. Methods: We consider possible correlations between the RVs and various stellar activity indicators accessible with CARMENES. We applied models within a Bayesian framework to determine whether a Keplerian model, a red-noise quasi-periodic model using a Gaussian process, or a mixed model would explain the observed data best. We also exclusively focus on spectral lines potentially associated with stellar activity. Results: The CARMENES RV data agree with the previously reported periodicity of 2.23d, correlate with some activity indicators, and exhibit chromaticity. However, when considering the entire RV data set, we find that a mixed model composed of a stable and a variable component performs best. Moreover, when recomputing the RVs using only spectral lines insensitive to activity, there appears to be some residual power at the period of interest. We therefore conclude that it is not possible to determinedly prove that there is no planet orbiting in synchronization with the stellar rotation given our data, current tools, machinery, and knowledge of how stellar activity affects RVs. We do rule out planets more massive than 27M_E (=0.084M_J). We also exclude any binary companion around AD Leo with Msini > 3-6M_J on orbital periods <14yr.