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Abstract We study the unitarized meson–baryon scattering amplitude at leading order in the strangeness $$S=-1$$ S = - 1 sector using time-ordered perturbation theory for a manifestly Lorentz-invariant formulation of chiral effective field theory. By solving the coupled-channel integral equations with the full off-shell dependence of the effective potential and applying subtractive renormalization, we analyze the renormalized scattering amplitudes and obtain the two-pole structure of the $$\Lambda (1405)$$ Λ ( 1405 ) resonance. We also point out the necessity of including higher-order terms.