The PADME experiment at the DA$\Phi$NE LINAC of the INFN Laboratori Nazionali di Frascati is designed to search for light, weakly coupled particles produced in positron--electron annihilation. The experiment features a positron beam impinging on a thin target and a detector optimized for the reconstruction of electromagnetic final states. Over multiple data-taking campaigns, PADME has collected data samples that enable both precision measurements of Standard Model processes and dedicated searches for physics beyond the SM.
\noindent This contribution focuses on the results obtained during the Run~III data-taking campaign, which implemented a fine-grained scan of the center-of-mass energy to search for a narrow resonance with mass around 17~MeV, as suggested by anomalies observed in nuclear transition experiments. No statistically significant signal is observed over the $(1.77\pm0.15)\sigma$ level, and upper limits are set on the coupling of a hypothetical 17~MeV vector boson to electrons. The Run~III results extend the experimental sensitivity of PADME to previously unexplored regions of parameter space.