  We introduce a non-Abelian discrete $\Delta_{27}$ family symmetry into the
recently proposed classes of Exceptional Supersymmetric Standard Model
($E_6$SSM) based on a broken $E_6$ Grand Unified Theory (GUT) in order to solve
the flavour problem in these models and in particular to account for
tri-bimaximal neutrino mixing. We consider both the minimal version of the
model (the ME$_6$SSM) with gauge coupling unification at the string scale and
the E$_6$SSM broken via the Pati-Salam chain with gauge coupling unification at
the conventional GUT scale. In both models there are low energy exotic colour
triplets with couplings suppressed by the symmetries of the model, including
the family symmetry. This leads to suppressed proton decay and long lived TeV
mass colour triplet states with striking signatures at the LHC. We also present
a dynamical solution to the $\mu'$ problem (where $\mu'$ is the mass of the
additional pair of electroweak doublets in the E$_6$SSM).
