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Abstract In this paper, we have studied the effect of Born–Infeld electrodynamics in holographic p-wave superconductors with massive vector condensation. We have analysed this model in the probe limit using a variational method known as the Stürm–Liouville eigenvalue approach. For this p-wave holographic superconductor model, we have calculated the critical temperature $$T_{c}$$ Tc as well as the value of the condensation operator for two different choices of $$m^{2}$$ m2 . We have also pointed out the similarities and dissimilarities between this model for $$m^{2} = 0$$ m2=0 and p-wave holographic superconductor model constructed out of Einstein–Yang–Mills theory. We have then computed the conductivity of these holographic superconductor models using a self-consistent approach and have shown that the DC conductivity diverges.