The exceptionally high luminosities of gamma-ray bursts (GRBs), gradually emerging as extremely useful probes of star formation, make them promising tools for exploration of the high-redshift Universe. Here we present a carefully selected sample of Swift GRBs, intended to estimate in an unbiased way the GRB mean redshift (z_mean), constraints on the fraction of high-redshift bursts and an upper limit on the fraction of heavily obscured afterglows. We find that z_mean = 2.8 and that at least 7% of GRBs originate at z > 5. In addition, consistent with pre-Swift observations, at most 20% of afterglows can be heavily obscured. The redshift distribution of the sample is qualitatively consistent with models where the GRB rate is proportional to the star formation rate in the Universe. We also report optical, near-infrared and X-ray observations of the afterglow of GRB 050814, which was seen to exhibit very red optical colours. By modelling its spectral energy distribution we find that z = 5.3 ± 0.3. The high mean redshift of GRBs and their wide redshift range clearly demonstrates their suitability as efficient probes of galaxies and the intergalactic medium over a significant fraction of the history of the Universe.