Context. High-precision and high-cadence photometric surveys such as Kepler or TESS are making huge progress not only in the detection of new extrasolar planets but also in the study of a great number of variable stars. This is the case for central stars of planetary nebulae (PNe), which have similarly benefited from the capabilities of these missions, increasing the number of known binary central stars and helping us to constrain the relationship between binarity and the complex morphologies of their host PNe. Aims. In this paper, we analyse the TESS light curves of a large sample of central stars of PNe with the aim of detecting signs of variability that may hint at the presence of short-period binary nuclei. This will have important implications in understanding PN formation evolution as well as the common envelope phase. Methods. We analysed 62 central stars of true, likely, or possible PNe and modelled the detected variability through an MCMC approach accounting for three effects: reflection, ellipsoidal modulations due to tidal forces, and the so-called Doppler beaming. Among the 62 central stars, only 38 are amenable for this study. The remaining 24 show large contamination from nearby sources preventing an optimal analysis. Also, eight targets are already known binary central stars, which we revisit here with the new high precision of the TESS data. Results. In addition to recovering the eight already known binaries in our sample, we find that 18 further central stars show clear signs of periodic variability in the TESS data, probably resulting from different physical effects compatible with the binary scenario. We propose them as new candidate binary central stars. We also discuss the origin of the detected variability in each particular case by using the TESS_localize algorithm. Finally, 12 targets show no or only weak evidence of variability at the sensitivity of TESS. Conclusions. Our study demonstrates the power of space-based photometric surveys in searching for close binary companions of central stars of PNe. Although our detections can only be catalogued as candidate binaries, we find a large percentage of possible stellar pairs associated with PNe, supporting the hypothesis that binarity plays a key role in shaping these celestial structures.