Context. This paper focuses on a class of galaxies characterised by an extremely low surface brightness: ultra-diffuse galaxies (UDGs). We used new integral-field (IF) spectroscopic data, obtained with the ESO Large Programme Looking into the faintEst WIth MUSE (LEWIS). It provides the first homogeneous IF spectroscopic survey performed by MUSE at the Very Large Telescope of a complete sample of UDGs and low-surface-brightness galaxies within a virial radius of 0.4 in the Hydra I cluster, according to the UDG abundance-halo mass relation. Aims. Our main goals are addressing the possible formation channels for this class of objects and investigating possible correlations of their observational properties, including the stacked (1D) and spatially resolved (2D) stellar kinematics. In particular, we derive the stellar velocity dispersion from the stacked spectrum integrated within the effective radius (σeff) and measure the velocity map of the galaxies in LEWIS. These quantities are used to estimate their dynamical mass (Mdyn). Methods. We extracted the 1D stacked spectrum inside the effective radius (Reff), which guarantees a high signal-to-noise ratio, to obtain an unbiased measure of σeff. To derive the spatially resolved stellar kinematics, we first applied the Voronoi tessellation algorithm to bin the spaxels in the datacube, and then we derived the stellar kinematics in each bin, following the same prescription as adopted for the 1D case. We extracted the velocity profiles along the galaxy major and minor axes and measured the semi-amplitude (ΔV) of the velocity curve. Results. We found that 7 out of 18 UDGs in LEWIS show a mild rotation (ΔV ∼ 25 ‑ 40 km s‑1), 5 lack evidence of any rotation, and the remaining 6 UDGs are unconstrained cases. This is the first large census of velocity profiles for UDGs. The UDGs in LEWIS are characterised by low values of σeff (≤30 km s‑1) on average, which is comparable with available values from the literature. Two objects show higher values of σeff (∼30 ‑ 40 km s‑1). These higher values might reasonably be due to the fast rotation observed in these galaxies, which affects the values of σeff. In the Faber-Jackson relation plane, we found a group of UDGs consistent with the relation within the error bars. Outliers of the Faber-Jackson relation are objects with a non-negligible rotation component. The UDGs and LSB galaxies in the LEWIS sample have a larger dark matter (DM) content on average than dwarf galaxies (Mdyn/LV, eff ∼ 10 ‑ 100 M⊙/L⊙) with a similar total luminosity. We do not find clear correlations between the derived structural properties and the local environment. Conclusions. By mapping the stellar kinematics for a homogenous sample of UDGs in a cluster environment, we found a significant rotation for many galaxies. Therefore, two classes of UDGs are found in the Hydra I cluster based on the stellar kinematics: rotating and non-rotating systems. This result, combined with the DM content and the upcoming analysis of the star formation history and globular cluster population, can help us to distinguish between the several formation scenarios proposed for UDGs.
