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, Na5Co15.5Te6O36. Upon cooling, magnetic susceptibility measurement reveals three successive magnetic transitions at T1 = 45.2 K, T2 = 35.4 K, and T3 ∼ 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 T3. Through the detailed characterizations, the magnetic field vs temperature phase diagram is established. A partially disordered antiferromagnetic state is proposed below T1, which becomes frozen below T3. Na5Co15.5Te6O36 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.