We prove the existence of a weakly dependent strictly stationary solution of the equation $ X_t=F(X_{t-1},X_{t-2},X_{t-3},...;ξ_t)$ called {\em chain with infinite memory}. Here the {\em innovations} $ξ_t$ constitute an independent and identically distributed sequence of random variables. The function $F$ takes values in some Banach space and satisfies a Lipschitz-type condition. We also study the interplay between the existence of moments and the rate of decay of the Lipschitz coefficients of the function $F$. With the help of the weak dependence properties, we derive Strong Laws of Large Number, a Central Limit Theorem and a Strong Invariance Principle.