In this paper, we consider the robust inverse scattering method for the Ablowitz–Ladik (AL) equation on the non-vanishing background, which can be used to deal with arbitrary-order poles on the branch points and spectral singularities in a unified way. The Darboux matrix is constructed with the aid of loop group method and considered within the framework of robust inverse scattering transform. Various soliton solutions are constructed without using the limit technique. These solutions include general soliton, breathers, as well as high order rogue wave solutions. •The robust inverse scattering transform of AL equation was derived.•The Riemann–Hilbert representation of Darboux matrix was given.•The distinct types of solitonic solutions were constructed.•The asymptotic analysis of multi-breathers was analyzed.•The sufficient conditions of highest peak for solitonic solutions were obtained.