In a recent paper we have classified fake projective planes. Natural higher dimensional generalization of these surfaces are arithmetic fake P_c^n-1, and arithmetic fake Gr_m,n. In this paper we show that arithmetic fake P_c^n-1 can exist only if n = 3, 5, and an arithmetic fake Gr_m,n can exist, with n > 3 odd, only if n = 5. Here we construct four distinct arithmetic fake P_c⁴, and four distinct fake arithmetic Gr_2,5. Furthermore, we use certain results and computations of [PY] to exhibit five irreducible arithmetic fake P_c². All these are connected smooth (complex projective) Shimura varieties.