We present echelle spectrophotometry of the blue compact dwarf galaxy NGC 5253 obtained with the VLT UVES. We have measured the intensities of a large number of permitted and forbidden emission lines in four zones of the central part of the galaxy. We detect faint C II and O II recombination lines, the first time that these are unambiguously detected in a dwarf starburst galaxy. The physical conditions of the ionized gas have been derived using a large number of different line intensity ratios. Chemical abundances of He, N, O, Ne, S, Cl, Ar, and Fe have been determined following standard methods. C++ and O++ abundances have been derived from pure recombination lines and are larger than those obtained from collisionally excited lines (from 0.30 to 0.40 dex for C++ and from 0.19 to 0.28 dex for O++). This result is consistent with a temperature fluctuation parameter (t2) between 0.050 and 0.072. We confirm previous results that indicate the presence of a localized N enrichment in certain zones of NGC 5253 and detect a possible slight He overabundance in the same zones. The enrichment pattern agrees with that expected for the pollution by the ejecta of Wolf-Rayet (W-R) stars. The amount of enriched material needed to produce the observed overabundance is consistent with the mass lost by the number of W-R stars estimated in the starbursts. We discuss the possible origin of the difference between abundances derived from recombination and collisionally excited lines (the so-called abundance discrepancy problem) in H II regions, finding that a recent hypothesis based on the delayed enrichment by SN ejecta inclusions seems not to explain the observed features. Based on observations collected at the European Southern Observatory, Chile, proposal ESO 70.C-0008(A).