We generalize the Picard–Lindelöf theorem on the unique solvability of initial value problems ẋ = f (t, x), x(t₀) = x₀, by replacing the sufficient classical Lipschitz condition of f with respect to x with a more general Lipschitz condition along hyperspaces of the (t, x)-space. A comparison with known results is provided and the generality of the new criterion is shown by an example.