We describe an approach to the dynamics of non-linear stochastic differential systems with finite memory using multiplicative cocycles in Hilbert space. We introduce the notion of hyperbolicity for stationary solutions of stochastic systems with memory. 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 ideas from infinite-dimensional multiplicative ergodic theory and interpolation arguments.