We show that the Lipschitz-free space F ( X ) \mathcal {F}(X) over a superreflexive Banach space X X has the property that every weakly precompact subset of F ( X ) \mathcal {F}(X) is relatively super weakly compact, showing that this space “behaves like L 1 L_1 ” in this context. As consequences we show that F ( X ) \mathcal {F}(X) enjoys the weak Banach-Saks property and that every subspace of F ( X ) \mathcal {F}(X) with nontrivial type is superreflexive. It follows from our results that weakly compact subsets of F ( X ) \mathcal {F}(X) are super weakly compact and hence have many strong properties. To prove the result, we use a modification of the proof of weak sequential completeness of F ( X ) \mathcal {F}(X) by Kochanek and Pernecká and an appropriate version of compact reduction in the spirit of Aliaga, Noûs, Petitjean and Procházka.