Viscoelastic fluids play a crucial role in a wide range of applications, spanning from polymer manufacturing to the pharmaceutical industry and biological research. They exhibit a behaviour which is intermediate between solids and fluids, as they do not possess a permanent reference state, but they keep a fading memory of prior deformations. Analytical solutions for free surface viscoelastic flows are often unavailable or excessively complex. As a result, over the past decades, numerical simulation techniques for viscoelastic flows have seen a growing interest both in research and in the industrial sector. This work represents the first example of the use of the so-called Particle Finite Element Method (PFEM) for the simulation of viscoelastic fluids. The PFEM is a Lagrangian approach combining standard FEM techniques with a remeshing strategy. It permits handling mesh distortion and easy tracking of the free surface. Even though originally developed for Newtonian flows, nowadays PFEM is also widely adopted to study non-Newtonian flows and large deformations problems in solid mechanics. Therefore, PFEM appears as a natural choice to study free surface viscoelastic problems. A PFEM implementation of the Oldroyd-B constitutive model coupled with the Navier-Stokes equations is proposed. It details the integration in time of the constitutive law and how to deal with the transfer of the internal variables, e.g., elastic strains, during remeshing events. The implementation is verified with free surface flow benchmarks like the impacting drop and the jet buckling. The proposed framework opens the way to complex viscoelastic simulations for real-world applications, such as 3D printing of polymers.