In this paper, we provide a more accurate description of the evolution of the magnetic flux redistribution during prestellar core collapse by including resistive terms in the magnetohydrodynamics (MHD) equations. We focus more particularly on the impact of ambipolar diffusion. We use the adaptive mesh refinement code RAMSES to carry out such calculations. The resistivities required to calculate the ambipolar diffusion terms were computed using a reduced chemical network of charged, neutral and grain species. The inclusion of ambipolar diffusion leads to the formation of a magnetic diffusion barrier in the vicinity of the core, preventing accumulation of magnetic flux in and around the core and amplification of the field above 0.1G. The mass and radius of the first Larson core remain similar between ideal and non-ideal MHD models. This diffusion plateau has crucial consequences on magnetic braking processes, allowing the formation of disk structures. Magnetically supported outflows launched in ideal MHD models are weakened when using non-ideal MHD. Contrary to ideal MHD misalignment between the initial rotation axis and the magnetic field direction does not significantly affect the results for a given mu, showing that the physical dissipation truly dominate over numerical diffusion. We demonstrate severe limits of the ideal MHD formalism, which yield unphysical behaviours in the long-term evolution of the system. This includes counter rotation inside the outflow, interchange instabilities, and flux redistribution triggered by numerical diffusion, none observed in non-ideal MHD. Disks with Keplerian velocity profiles form in all our non-ideal MHD simulations, with final mass and size which depend on the initial magnetisation. This ranges from a few 0.01 solar masses and 20-30 au for the most magnetised case (mu=2) to 0.2 solar masses and 40-80 au for a lower magnetisation (mu=5).
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