  We investigate the near-horizon geometry of supersymmetric extremal black
holes in 6-dimensional gauged $N=(1,0)$ supergravity. We solve the Killing
spinor equations along the lightcone and establish the independent differential
and algebraic conditions which are given as the naive restriction of the KSEs
on ${\cal S}$. By analyzing the global properties of the Killing spinors, we
prove that the near-horizon geometries undergo a (super)symmetry enhancement.
This follows from generalized Lichnerowicz-type theorems for the zero modes of
the Dirac operator and an index theory argument. We also prove that horizons
with non-trivial fluxes admit an $\mathfrak{sl}(2,\mathbb{R})$ symmetry group.
