  We investigate the fragmentation instability of hairy black holes in the
theory with a Gauss$-$Bonnet(GB) term in asymptotically flat spacetime. Our
approach is through the non-perturbative fragmentation instability. By this
approach, we investigate whether the initial black hole can be broken into two
black holes by comparing the entropy of the initial black hole with the sum of
those of two fragmented black holes. The relation between the black hole
instability and the GB coupling with dilaton hair are presented. We describe
the phase diagrams with respect to the mass of the black hole solutions and
coupling constants. We find that a perturbatively stable black hole can be
unstable under fragmentation.
