We consider a variety of lattice spin systems (including Ising, Potts and XY models) on $\mathbb{Z}^d$ with long-range interactions of the form $J_x = ψ(x) e^{-|x|}$, where $ψ(x) = e^{\mathsf{o}(|x|)}$ and $|\cdot|$ is an arbitrary norm. We characterize explicitly the prefactors $ψ$ that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature $β$, magnetic field $h$, etc). Our results apply in any dimension. As an interesting particular case, we prove that, in one-dimensional systems, the correlation length is non-analytic whenever $ψ$ is summable, in sharp contrast to the well-known analytic behavior of all standard thermodynamic quantities. We also point out that this non-analyticity, when present, also manifests itself in a qualitative change of behavior of the 2-point function. In particular, we relate the lack of analyticity of the correlation length to the failure of the mass gap condition in the Ornstein--Zernike theory of correlations.