We show that ultraviolet divergences found in fermionic Green functions of massless QED₂ have an essentially non-perturbative nature. We investigate their origin both in the gauge invariant formalism (where we introduce Wess-Zumino fields to restore quantum gauge invariance) and in the gauge non-invariant formalism, mapping two different but equivalent mechanisms responsible for their appearance. We find the same results in both approaches, what contradicts a previous work of Jian-Ge, Qing-Hai and Yao-Yang, that found no divergences in the chiral Schwinger model considered in the gauge invariant formalism.