3D reconstruction of highly deformable surfaces (e.g. cloths) from monocular RGB videos is a challenging problem, and no solution provides a consistent and accurate recovery of fine-grained surface details. To account for the ill-posed nature of the setting, existing methods use deformation models with statistical, neural, or physical priors. They also predominantly rely on nonadaptive discrete surface representations (e.g. polygonal meshes), perform frame-by-frame optimisation leading to error propagation, and suffer from poor gradients of the mesh-based differentiable renderers. Consequently, fine surface details such as cloth wrinkles are often not recovered with the desired accuracy. In response to these limitations, we propose Thin-Shell-SfT, a new method for non-rigid 3D tracking that represents a surface as an implicit and continuous spatiotemporal neural field. We incorporate continuous thin shell simulation based on the Kirchhoff-Love model for spatial regularisation, which starkly contrasts the discretised alternatives of earlier works. Lastly, we leverage 3D Gaussian splatting to differentiably render the surface into image space and optimise the deformations based on analysis-by-synthesis principles. Our Thin-Shell-SfT method outperforms prior work qualitatively and quantitatively thanks to our continuous surface formulation in conjunction with a specially tailored simulation prior and joint space-time optimisation.