We present a continuous-time probabilistic approach for estimating the chirp signal and its instantaneous frequency function when the true forms of these functions are not accessible. Our model represents these functions by non-linearly cascaded Gaussian processes represented as non-linear stochastic differential equations. The posterior distribution of the functions is then estimated with stochastic filters and smoothers. We compute a (posterior) Cramér--Rao lower bound for the Gaussian process model, and derive a theoretical upper bound for the estimation error in the mean squared sense. The experiments show that the proposed method outperforms a number of state-of-the-art methods on a synthetic data. We also show that the method works out-of-the-box for two real-world datasets.