A search for charginos with masses close to the mass of the lightest neutralino is reported, based on the data collected with the DELPHI detector at LEP from 1995 to 1997 at centre-of-mass energies between 130 and 183 GeV. The signature of a photon at high transverse momentum radiated from the initial state reduces the two-photon background to acceptable rates, thus making the mass differences between a few hundred MeV/c2 and 3 GeV/c2 detectable. In very nearly degenerate scenarios, the lifetime of the chargino can be large enough to produce either visible secondary vertices or decays outside the detector; therefore, quasi-stable heavy charged particles and displaced decay vertices were also searched for. No excess of events with respect to the Standard Model expectations was observed, and limits in the plane of chargino-neutralino mass difference versus chargino mass are given.