Very weak Bernoulli processes with values in a separable metric space are introduced. An estimate for the Prohorov distance in the central limit theorem is obtained. This estimate is used to establish a strong (almost sure) approximation of the partial sums of a very weak Bernoulli process by a Brownian motion where the error term is of the order O(t1/2−γ). The proofs are based on a new version of the Berkes-Philipp approximation theorem.