  Extremal black holes in general dimensions are well known to contain an
AdS$_2$ factor in their near-horizon geometries. If the extremal limit is taken
in conjunction with a specific vanishing horizon limit, the so-called Extremal
Vanishing Horizon (EVH) limit, the AdS$_2$ factor lifts to a locally AdS$_3$
factor with a pinching angular direction. In this paper, we study the EVH limit
of asymptotically AdS black holes which preserve some supersymmetry. The
primary example we consider is the 1/16$^{\rm th}$ BPS asymptotically AdS$_5$
black hole, whose EVH limit has an AdS$_3$ factor in its near-horizon geometry.
We also consider the near-EVH limit of this black hole, in which the
near-horizon geometry instead contains an extremal BTZ factor. We employ recent
results on the large-$N$ limit of the superconformal index of the dual CFT$_4$
to understand the emergence of a CFT$_2$ in the IR of the CFT$_4$, which is the
field theory dual to the emergence of the locally AdS$_3$ factor in the
near-horizon geometry. In particular, we show that the inverse Laplace
transform of the superconformal index, yielding the black hole entropy, becomes
equivalent to the derivation of a Cardy formula for the dual CFT$_2$. Finally,
we examine the EVH limit of supersymmetric black holes in other dimensions.
