In superconductors, electrons are paired and condensed into the ground state. An impurity can break the electron pairs into quasiparticles with energy states inside the superconducting gap. The characteristics of such in-gap states reflect accordingly the properties of the superconducting ground state. A zero-energy in-gap state is particularly noteworthy, because it can be the consequence of non-trivial pairing symmetry or topology. Here we use scanning tunnelling microscopy/spectroscopy to demonstrate that an isotropic zero-energy bound state with a decay length of ∼10 Å emerges at each interstitial iron impurity in superconducting Fe(Te,Se). More noticeably, this zero-energy bound state is robust against a magnetic field up to 8 T, as well as perturbations by neighbouring impurities. Such a spectroscopic feature has no natural explanation in terms of impurity states in superconductors with s-wave symmetry, but bears all the characteristics of the Majorana bound state proposed for topological superconductors, indicating that the superconducting state and the scattering mechanism of the interstitial iron impurities in Fe(Te,Se) are highly unconventional.