As a new antiperovskite nitride, ZnFe₃N was synthesized and characterized by almost completely substituting iron atoms at corner positions of γ′-Fe₄N. The magnetic interactions of the system with the space group Pmm are fully investigated. The critical behavior was investigated based on the measured magnetic data around the ferromagnetic phase transition temperature. In this work, the values of critical exponents (β, γ and δ) were obtained systematically using the Kouvel–Fisher method in the critical region. The Widom scaling law (δ = 1 + γ^β−1) and the scaling equation (m = f_±(h)) were used to reveal the reliability of these values. The values of the critical exponents (β = 0.325, γ = 1.228, and δ = 4.778) are different from those predicted by the three-dimensional (3D) Heisenberg model and mean-field model, and are very close to those of the 3D-Ising model. Combined with ESR analysis, the spin clusters induced by changes in chemical bonds are considered to be the cause for the existence of an anisotropic short-range ordered state in this ferromagnetic system.