You can't find a reference because it's false. Rademacher's theorem (Lebesgue-almost everywhere differentiability) is the best one can do.
In fact, for every Lebesgue-null set $E \subset \mathbf{R}$, you can construct a Lipschitz function $f: \mathbf{R} \to \mathbf{R}$ that is not differentiable at any point of $E$. [ACP10]
The reference for this is Theorem 1.1 in... the same paper you linked in your question. The result is on page 1.
[ACP10] G. Alberti, M. Csörnyei, and D. Preiss. Differentiability of lipschitz functions, structure of null sets, and other problems. In Proceedings of the ICM 2010, pages 1379-1394.