Numerical simulations of real-world systems often involve complex geometries, either in the form of boundaries, phase interfaces, or distributed sources. Accurate and efficient representation and treatment of complex geometries requires versatile geometric computing frameworks. These can then also be used to evolve deformable shapes and derive differential-geometric quantities, such as surface normals and curvatures. So far in Lagrangian particle methods, colorfield-based approaches have been most widely used to compute surface normals and mean curvatures. They are robust and simple. However, using a binary identifier function limits the overall accuracy of the computed geometric quantities. To improve accuracy, level-set methods use higher-order distance functions to implicitly describe geometries and handle changes in topology. However, most level-set methods still use a regular mesh for level-set redistancing and numerical consistency. This requires particle-mesh interpolation of the distance function, which, as we demonstrate here, again limits the accuracy of geometrical quantities. To overcome the need for remeshing, we extend the work of Saye [1] to a purely Lagrangian particle level-set method that performs well on irregular particle distributions in a narrow band. Our method is based on high-degree regression using Newton-Lagrange polynomials on unisolvent nodes [2], which enables high-order computation of surface normals and curvatures. We show high-order convergence in benchmarks on basic geometries and test the robustness of the method for a spiralling vortex. Further, we present applications to multi-phase flow simulations using Smoothed Particle Hydrodynamics (SPH) to simulate an oscillating droplet. We compare the results of our formulation with those computed using a classic colorfield approach [3]. [1] R. Saye, High-order methods for computing distances to implicitly defined surfaces, Communications in Applied Mathematics and Computational Science, (2014) [2] M. Hecht, K. Gonciarz, J. Michelfeit, V. Sivkin, and I. F. Sbalzarini, Multivariate interpolation in unisolvent nodes–lifting the curse of dimensionality, arXiv preprint, (2020) [3] J. P. Morris, Simulating surface tension with smoothed particle hydrodynamics, International journal for numerical methods in fluids, (2000)