State-space models (SSM) are often used for analyzing complex ecological processes that are not observed directly, such as marine animal movement. When outliers are present in the measurements, special care is needed in the analysis to obtain reliable location and process estimates. Here we recommend using the Laplace approximation combined with automatic differentiation (as implemented in the novel R package Template Model Builder; TMB) for the fast fitting of continuous-time multivariate non-Gaussian SSMs. Through Argos satellite tracking data, we demonstrate that the use of continuous-time t -distributed measurement errors for error-prone data is more robust to outliers and improves the location estimation compared to using discretized-time t -distributed errors (implemented with a Gibbs sampler) or using continuous-time Gaussian errors (as with the Kalman filter). Using TMB, we are able to estimate additional parameters compared to previous methods, all without requiring a substantial increase in computational time. The model implementation is made available through the R package argosTrack.