We state and prove a Local Stable Manifold Theorem for nonlinear stochastic differential systems with finite memory (viz. stochastic functional differential equations (sfde's)). We introduce the notion of hyperbolicity for stationary solutions of sfde's. We then establish the existence of smooth stable and unstable manifolds in a neighborhood of a hyperbolic stationary solution. The stable and unstable manifolds are stationary and asymptotically invariant under the stochastic semiflow. The proof uses infinite-dimensional multiplicative ergodic theory techniques and interpolation arguments.