We use heavy baryon chiral perturbation theory to evaluate the two-photon exchange corrections to the low-energy elastic lepton-proton scattering at next-to-leading order accuracy, i.e., ${mathcal O}(alpha, M^{-1})$, including a non-zero lepton mass. We consider the elastic proton intermediate state in the two-photon exchange together in the soft photon approximation. The infrared singular contributions are projected out using dimensional regularization. The resulting infrared singularity-free two-photon exchange contribution is in good numerical agreement with existing predictions based on standard diagrammatic soft photon approximation evaluations.
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