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We numerically investigate the holographic paramagnetism–ferromagnetism phase transition in the 4-dimensional Lifshitz spacetime in the presence of three kinds of typical Born–Infeld-like nonlinear electrodynamics. Concretely, in the probe limit, we thoroughly discuss the effects of the nonlinear parameter b and the dynamical exponent z on the critical temperature, magnetic moment and hysteresis loop. The results show that the exponential form of nonlinear electrodynamics correction makes the critical temperature smaller and the magnetic moment harder to form with the absent external field for a constant nonlinear parameter b comparing it with the logarithmic form of nonlinear electrodynamics and the Born–Infeld nonlinear electrodynamics, especially for the case of larger dynamical exponent z. Moreover, the increase of nonlinear parameter b (for the fixed z) or dynamical exponent z (for the fixed b) will result in extending the period of the external magnetic field. Particularly, the effect of the exponential form of nonlinear electrodynamics on the periodicity of hysteresis loop is more noteworthy.