Context. Disc-halo decompositions z = 1 − 2 star-forming galaxies (SFGs) at z > 1 are often limited to massive galaxies (M⋆ > 1010 M⊙) and rely on either deep integral field spectroscopy data or stacking analyses.Aims. We present a study of the dark-matter (DM) content of nine z ≈ 1 SFGs selected among the brightest [O II] emitters in the deepest Multi-Unit Spectrograph Explorer (MUSE) field to date, namely the 140 h MUSE Extremely Deep Field. These SFGs have low stellar masses, ranging from 108.5 to 1010.5 M⊙.Methods. We analyzed the kinematics with a 3D modeling approach, which allowed us to measure individual rotation curves to ≈3 times the half-light radius Re. We performed disk-halo decompositions on their [O II] emission line with a 3D parametric model. The disk-halo decomposition includes a stellar, DM, gas, and occasionally a bulge component. The DM component primarily uses the generalized α, β, γ profile or a Navarro-Frenk-White profile.Results. The disk stellar masses M⋆ obtained from the [O II] disk-halo decomposition agree with the values inferred from the spectral energy distributions. While the rotation curves show diverse shapes, ranging from rising to declining at large radii, the DM fractions within the half-light radius fDM(< Re) are found to be 60% to 95%, extending to lower masses (densities) recent results who found low DM fractions in SFGs with M⋆ > 1010 M⊙. The DM halos show constant surface densities of ∼100 M⊙ pc−2. For isolated galaxies, half of the sample shows a strong preference for cored over cuspy DM profiles. The presence of DM cores appears to be related to galaxies with low stellar-to-halo mass ratio, log M⋆/Mvir ≈ −2.5. In addition, the cuspiness of the DM profiles is found to be a strong function of the recent star-formation activity.Conclusions. We measured the properties of DM halos on scales from 1 to 15 kpc, put constraints on the z > 0 cvir − Mvir scaling relation, and unveiled the cored nature of DM halos in some z ≃ 1 SFGs. These results support feedback-induced core formation in the cold dark matter context.