Using a data set of 2.93 fb⁻¹ taken at a center-of-mass energy √s = 3.773 GeV with the BESIII detector at the BEPCII collider, we perform a search for an extra U(1) gauge boson, also denoted as a dark photon. We examine the initial state radiation reactions e⁺e⁻ → e⁺e⁻γ(ISR) and e⁺e⁻ → μ⁺μ⁻γ(ISR) for this search, where the dark photon would appear as an enhancement in the invariant mass distribution of the leptonic pairs. We observe no obvious enhancement in the mass range between 1.5 and 3.4 GeV/c² and set a 90% confidence level upper limit on the mixing strength of the dark photon and the Standard Model photon. We obtain a competitive limit in the tested mass range.