Newtonian fluids have no resistance to deviatoric stresses and can deform indefinitely. That is why most researchers adopt the Eulerian description, having as main variables the velocities in the current configuration, with indeformable spatial domain. However, Eulerian description cannot be directly applied to moving boundary flows, such as free-surface flows, multiple fluid flows and fluid-structure interaction. For some free-surface flows, a Lagrangian description can be more suitable, specially if the fluid volume is enclosed and presents only finite deformations and it makes sense to consider current nodal positions as main variables. In this work, we present a position-based space-time finite element formulation for incompressible Newtonian fluid flows using total a Lagrangian description applied to free-surface flows with finite strains. Such approach uses mixed position-pressure finite elements, with Petrov-Galerking pressure stabilization. The discretization is structured in time direction, so that the space-time shape functions are given as a tensor product of the linear space shape-functions set with the quadratic shape functions set in time direction. Thus, the space-time domain is composed of space-time slabs, that can be solved progressively, with velocities and positions in the previous slab being imposed as initial conditions on the current space-time slab. The resulting formulation is tested by numerical examples, revealing to be robust and efficient for simulating free-surface incompressible flows, like sloshing and wave propagation.