Context. Due to their relatively low stellar mass content and diffuse nature, the evolution of dwarf galaxies can be strongly affected by their environment. Analyzing the properties of the dwarf galaxies over a wide range of luminosities, sizes, morphological types, and environments, we can obtain insights about their evolution. At ∼50 Mpc, the Hydra I cluster of galaxies is among the closest cluster in the z ≃ 0 Universe, and an ideal environment to study dwarf galaxy properties in a cluster environment. Aims. We exploit deep imaging data of the Hydra I cluster to construct a new photometric catalog of dwarf galaxies in the cluster core, which is then used to derive properties of the Hydra I cluster dwarf galaxy population as well as to compare it with other clusters. Moreover, we investigate the dependency of dwarf galaxy properties on their surrounding environment. Methods. The new wide-field g- and r-band images of the Hydra I cluster obtained with the OmegaCAM camera on the VLT Survey Telescope (VST) in the context of the VST Early-type GAlaxy Survey (VEGAS) were used to study the dwarf galaxy population in the Hydra I cluster core down to r-band magnitude M_r = −11.5 mag. We used an automatic detection tool to identify dwarf galaxies from a ∼1 deg² field centered on the Hydra I core, covering almost half of the cluster virial radius. The photometric pipeline was used to estimate the principal photometric parameters for all targets. Scaling relations and visual inspection were used to assess the cluster membership and construct a new dwarf galaxy catalog. Finally, based on the new catalog, we studied the structural (Sérsic index n, effective radius R_e, and axis ratio) and photometric (colors and surface brightness) properties of the dwarf galaxies, also investigating how they vary as a function of clustercentric distance. Results. The new Hydra I dwarf catalog contains 317 galaxies with a luminosity between −18.5 < M_r < −11.5 mag, a semi-major axis larger than ∼200 pc (a = 0.84″), of which 202 are new detections, and previously unknown dwarf galaxies in the Hydra I central region. We estimate that our detection efficiency reaches 50% at the limiting magnitude M_r = −11.5 mag, and at the mean effective surface brightness μ̄_e,r = 26.5 mag arcsec⁻². We present the standard scaling relations for dwarf galaxies, which are color-magnitude, size-luminosity, and Sérsic n-magnitude relations, and compare them with other nearby clusters. We find that there are no observational differences for dwarfs scaling relations in clusters of different sizes. We study the spatial distribution of galaxies, finding evidence for the presence of substructures within half the virial radius. We also find that mid- and high-luminosity dwarfs (M_r < −14.5 mag) become, on average, redder toward the cluster center, and that they have a mild increase in R_e with increasing clustercentric distance, similar to what is observed for the Fornax cluster. No clear clustercentric trends are reported for surface brightness and Sérsic index. Considering galaxies in the same magnitude bins, we find that for high and mid-luminosity dwarfs (M_r < −13.5 mag), the g − r color is redder for the brighter surface brightness and higher Sérsic n index objects. This finding is consistent with the effects of harassment and/or partial gas stripping.