This paper presents a novel Learning-based Model Predictive Contouring Control (L-MPCC) algorithm for evasive manoeuvres at the limit of handling. The algorithm uses the Student-t Process (STP) to minimise model mismatches and uncertainties online. The proposed STP captures the mismatches between the prediction model and the measured lateral tyre forces and yaw rate. The mismatches correspond to the posterior means provided to the prediction model to improve its accuracy. Simultaneously, the posterior covariances are propagated to the vehicle lateral velocity and yaw rate along the prediction horizon. The STP posterior covariance directly depends on the variance of observed data, so its variance is more significant when the online measurements differ from the recorded ones in the training set and smaller in the opposite case. Thus, these covariances can be utilised in the L-MPCC's cost function to minimise the vehicle state uncertainties. In a high-fidelity simulation environment, we demonstrate that the proposed L-MPCC can successfully avoid obstacles, keeping the vehicle stable while driving a double lane change manoeuvre at a higher velocity than an MPCC without STP. Furthermore, the proposed controller yields a significantly lower peak sideslip angle, improving the vehicle's manoeuvrability compared to an L-MPCC with a Gaussian Process.