We propose a novel approach, based on principal components analysis, to the use of gamma-ray bursts (GRBs) as probes of cosmic star formation history (SFH) up to very high redshifts. The main advantage of such approach is to avoid the necessity of assuming an ad hoc parametrization of the SFH. We first validate the method by reconstructing a known SFH from Monte Carlo generated mock data. We then apply the method to the most recent Swift data of GRBs with known redshift and compare it against the SFH obtained by independent methods. The main conclusion is that the level of star formation activity at z≈ 9.4 could have been already as high as the present-day one (≈0.01 M⊙ yr−1 Mpc−3). This is a factor 3–5 times higher than deduced from high-z galaxy searches through drop-out techniques. If true, this might alleviate the longstanding problem of a photon-starving reionization; it might also indicate that galaxies accounting for most of the star formation activity at high redshift go undetected by even the most deep searches.