We present exact calculations of the zero-temperature partition function of the $q$-state Potts antiferromagnet on arbitrarily long strips of the square, triangular, and kagom\'e lattices with width $L_y=2$ or 3 vertices and with periodic longitudinal boundary conditions. From these, in the limit of infinite length, we obtain the exact ground-state entropy $S_0=k_B \ln W$. These results are of interest since this model exhibits nonzero ground state entropy $S_0 > 0$ for sufficiently large $q$ and hence is an exception to the third law of thermodynamics. We also include results for homeomorphic expansions of the square lattice strip. The analytic properties of $W(q)$ are determined and related to zeros of the chromatic polynomial for long finite strips.