We define a new class of Gaussian processes on compact metric graphs such as street or river networks. The proposed models, the Whittle--Matérn fields, are defined via a fractional stochastic differential equation on the compact metric graph and are a natural extension of Gaussian fields with Matérn covariance functions on Euclidean domains to the non-Euclidean metric graph setting. Existence of the processes, as well as some of their main properties, such as sample path regularity are derived. The model class in particular contains differentiable processes. To the best of our knowledge, this is the first construction of a differentiable Gaussian process on general compact metric graphs. Further, we prove an intrinsic property of these processes: that they do not change upon addition or removal of vertices with degree two. Finally, we obtain Karhunen--Loève expansions of the processes, provide numerical experiments, and compare them to Gaussian processes with isotropic covariance functions.