A novel estimator for the identification of continuous-time linear time-varying systems is presented in this paper. The estimator uses kernel-based regression to identify the time-varying coefficients of a linear ordinary differential equation, based on noisy samples of the input and output signals. The estimator adopts a mixed time- and frequency-domain formulation, which allows it to be formulated as the solution of a set of algebraic equations, without relying on finite differences to approximate the time derivatives. Since a kernel-based approach is used, the model complexity selection of the time-varying parameters is formulated as an optimisation problem with continuous variables. Variance and bias expressions of the estimate are derived and validated on a simulation example. Also, it is shown that, in highly noisy environments, the proposed kernel-based estimator provides more reliable results than an Oracle'-based estimator which is deprived of regularisation.