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Data-driven discovery of "hidden physics" -i.e., machine learning of differential equation models underlying observed data -has recently been approached by embedding the discovery problem into a Gaussian Process regression of spatial data, treating and discovering unknown equation parameters as hyperparameters of a "physics informed" Gaussian Process kernel. This kernel includes the parametrized differential operators applied to a prior covariance kernel. We extend this framework to the data-driven discovery of linear space-fractional differential equations. The methodology is compatible with a wide variety of space-fractional operators in R d and stationary covariance kernels, including the Matérn class, and allows for optimizing the Matérn parameter during training. Since fractional derivatives are typically not given by closed-form analytic expressions, the main challenges to be addressed are a user-friendly, general way to set up fractional-order derivatives of covariance kernels, together with feasible and robust numerical methods for such implementations. Making use of the simple Fourier-space representation of space-fractional derivatives in R d , we provide a unified set of integral formulas for the resulting Gaussian Process kernels. The shift property of the Fourier transform results in formulas involving d-dimensional integrals that can be efficiently treated using generalized Gauss-Laguerre quadrature.The implementation of fractional derivatives has several benefits. First, the method allows for discovering models involving fractional-order PDEs for systems characterized by heavy tails or anomalous diffusion, while bypassing the analytical difficulty of fractional calculus. Data sets exhibiting such features are of increasing prevalence in physical and financial domains. Second, a single fractional-order archetype allows for a derivative term of arbitrary order to be learned, with the order itself being a parameter in the regression. As a result, even when used for discovering integer-order equations, the proposed method has several benefits compared to previous works on data-driven discovery of differential equations; the user is not required to assume a "dictionary" of derivatives of various orders, and directly controls the parsimony of the models being discovered. We illustrate our method on several examples, including fractional-order interpolation of advection-diffusion and modeling relative stock performance in the S&P 500 with α-stable motion via a fractional diffusion equation.