Cleaves proteins, imported into the mitochondrion, to their mature size. While most mitochondrial precursor proteins are processed to the mature form in one step by mitochondrial processing peptidase (MPP), the sequential cleavage by MIP of an octapeptide after initial processing by MPP is a required step for a subgroup of nuclear-encoded precursor proteins destined for the matrix or the inner membrane (By similarity). Release of an N-terminal octapeptide as second stage of processing of some proteins imported into the mitochondrion. Binds 1 zinc ion. Belongs to the peptidase M3 family.