We present the first results of a project, Lyman-break galaxies Stellar populations and Dynamics (LSD), aimed at obtaining spatially resolved, near-infrared (IR) spectroscopy of a complete sample of Lyman-break galaxies at z ~ 3. Deep observations with adaptive optics resulted in the detection of the main optical lines, such as [OII] lambda3727, Hbeta and [OIII] lambda5007, which are used to study sizes, star formation rates (SFRs), morphologies, gas-phase metallicities, gas fractions and effective yields. Optical, near-IR and Spitzer/Infrared Array Camera photometry are used to measure stellar mass. We obtain that morphologies are usually complex, with the presence of several peaks of emissions and companions that are not detected in broad-band images. Typical metallicities are 10-50 per cent solar, with a strong evolution of the mass-metallicity relation from lower redshifts. Stellar masses, gas fraction and evolutionary stages vary significantly among the galaxies, with less massive galaxies showing larger fractions of gas. In contrast with observations in the local universe, effective yields decrease with stellar mass and reach solar values at the low-mass end of the sample. This effect can be reproduced by gas infall with rates of the order of the SFRs. Outflows are present but are not needed to explain the mass-metallicity relation. We conclude that a large fraction of these galaxies is actively creating stars after major episodes of gas infall or merging.