We define certain spaces of Banach-valued measures called Lipschitz measures. When the Banach space is a dual space X * , these spaces can be identified with the duals of the atomic vector-valued Hardy spaces H X p ( ℝ n ) , 0 < p < 1 . We also prove that all these measures have Lipschitz densities. This implies that for every real Banach space X and 0 < p < 1 , the dual H X p ( ℝ n ) ∗ can be identified with a space of Lipschitz functions with values in X * .