  $S$ = 1/2 Ising kagome antiferromagnets (KAFs) are rare in reality. In this
work, we report the magnetic ground state and field-induced transitions of an
$S$ = 1/2 Ising antiferromagnet with a stacked kagome geometry,
Na$_5$Co$_{15.5}$Te$_6$O$_{36}$. Upon cooling, magnetic susceptibility
measurement reveals three successive magnetic transitions at $T_1$ = 45.2 K,
$T_2$ = 35.4 K, and $T_3 \sim$ 20 K. When the magnetic field is applied along
the Ising $c$ axis, a strikingly anomalous initial magnetization lying outside
the hysteresis loop is observed below $T_3$. Through the detailed
characterizations, the magnetic field $vs$ temperature phase diagram is
established. A partially disordered antiferromagnetic state is proposed below
$T_1$, which becomes frozen below $T_3$. Na$_5$Co$_{15.5}$Te$_6$O$_{36}$ is
therefore a promising candidate for $S$ = 1/2 Ising KAFs, and this unique
composite structure may shed light on the exploratory path for $S$ = 1/2 Ising
KAFs.
