We consider a variety of lattice spin systems (including Ising, Potts and XY models) on Z^d with long-range interactions of the form J_x = psi(x) exp(-|x|), where psi(x) = exp(o(|x|)) and |·| is an arbitrary norm. We characterize explicitly the prefactors psi that give rise to a correlation length that is not analytic in the relevant external parameter(s) (inverse temperature ß, 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 psi 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.