Cylinders in del Pezzo fibrations

We show that a del Pezzo fibration pi: V -> W of degree d contains a vertical open cylinder, that is, an open subset whose intersection with the generic fiber of pi is isomorphic to Z x A (K) (1) for some quasi-projective variety Z defined over the function field K of W, if and only if d >= 5 and pi: V -> W admits a rational section. We also construct twisted cylinders in total spaces of threefold del Pezzo fibrations pi: V -> P (1) of degree d <= 4.