We define a locally convex space E to have the Josefson–Nissenzweig property (JNP) if the identity map (E ′ , σ(E ′ , E)) → (E ′ , β * (E ′ , E)) is not sequentially continuous. By the classical Josefson–Nissenzweig theorem, every infinite-dimensional Banach space has the JNP. A characterization of locally convex spaces with the JNP is given. We thoroughly study the JNP in various function spaces. Among other results we show that for a Tychonoff space X, the function space C p (X) has the JNP iff there is a weak * null-sequence (µ n) n∈ω of finitely supported sign-measures on X with unit norm. However, for every Tychonoff space X, neither the space B 1 (X) of Baire-1 functions on X nor the free locally convex space L(X) over X has the JNP.