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We present a catalogue of 4499 groups and clusters of galaxies from the first data release of the multi-filter (5 broad, 7 narrow) Southern Photometric Local Universe Survey (S-PLUS). These groups and clusters are distributed over 273 deg$^2$ in the Stripe 82 region. They are found using the PzWav algorithm, which identifies peaks in galaxy density maps that have been smoothed by a cluster scale difference-of-Gaussians kernel to isolate clusters and groups. Using a simulation-based mock catalogue, we estimate the purity and completeness of cluster detections: at S/N>3.3 we define a catalogue that is 80% pure and complete in the redshift range 0.1<z<0.4, for clusters with $M_{200} > 10^{14}$ M$_\odot$. We also assessed the accuracy of the catalogue in terms of central positions and redshifts, finding scatter of $σ_R=12$ kpc and $σ_z=8.8 \times 10^{-3}$, respectively. Moreover, less than 1% of the sample suffers from fragmentation or overmerging. The S-PLUS cluster catalogue recovers ~80% of all known X-ray and Sunyaev-Zel'dovich selected clusters in this field. This fraction is very close to the estimated completeness, thus validating the mock data analysis and paving an efficient way to find new groups and clusters of galaxies using data from the ongoing S-PLUS project. When complete, S-PLUS will have surveyed 9300 deg$^{2}$ of the sky, representing the widest uninterrupted areas with narrow-through-broad multi-band photometry for cluster follow-up studies.