  The Eddington-inspired-Born-Infeld (EiBI) model is reformulated within the
mimetic approach. In the presence of a mimetic field, the model contains
non-trivial vacuum solutions which could be free of spacetime singularity
because of the Born-Infeld nature of the theory. We study a realistic
primordial vacuum universe and prove the existence of regular solutions, such
as primordial inflationary solutions of de Sitter type or bouncing solutions.
Besides, the linear instabilities present in the EiBI model are found to be
avoidable for some interesting bouncing solutions in which the physical metric
as well as the auxiliary metric are regular at the background level.
