We report on our efforts to test the Einstein Equivalence Principle by measuring the gravitational redshift with the VLBI spacecraft RadioAstron, in an eccentric orbit around Earth with geocentric distances as small as ∼7000 km and up to 350,000 km. The spacecraft and its ground stations are each equipped with stable hydrogen maser frequency standards, and measurements of the redshifted downlink carrier frequencies were obtained at both 8.4 and 15 GHz between 2012 and 2017. Over the course of the ∼9 d orbit, the gravitational redshift between the spacecraft and the ground stations varies between 6.8×10-10 and 0.6×10-10. Since the clock offset between the masers is difficult to estimate independently of the gravitational redshift, only the variation of the gravitational redshift is considered for this analysis. We obtain a preliminary estimate of the fractional deviation of the gravitational redshift from prediction of [Formula preseted] with the systematic uncertainty likely being dominated by unmodelled effects including the error in accounting for the non-relativistic Doppler shift. This result is consistent with zero within the uncertainties. For the first time, the gravitational redshift has been probed over such large distances in the vicinity of Earth. About three orders of magnitude more accurate estimates may be possible with RadioAstron using existing data from dedicated interleaved observations combining uplink and downlink modes of operation.