We show that the recently introduced classical Arnol'd cat map lattice field theories, which are chaotic, are exponentially mixing to all orders. Their mixing times are well-defined and are expressed in terms of the Lyapunov exponents, more precisely by the combination that defines the inverse of the Kolmogorov-Sinai entropy of these systems. We prove by an explicit recursive construction of correlation functions, that these exhibit $l-$fold mixing for any l= 3,4,5... This computation is relevant for Rokhlin's conjecture, which states that 2-fold mixing induces l-fold mixing for any l>2. Our results show that 2-fold exponential mixing, while being necessary for any $l-$fold mixing to hold it is nevertheless not sufficient for Arnol'd cat map lattice field theories. The correspondence principle implies that these mixing times, also, control the scrambling of the underlying quantum system for short times.