Let Π be a thick polar space of rank n ≥ 3. Pick a hyperplane F of Π and Η of Π. Define the elements of a biaffine polar space Γ to be those elements of Π which are not contained in F, or dually in Η. We show that Γ is a simply connected geometry, except for a few small exceptions for Π. We give a construction that leads to flag-transitive examples.