An \({\mathbb A}^r_{k}\)-cylinder in a normal algebraic variety (defined over \(k\)) is a Zariski open subset \(U\) isomorphic to a product \(Z \times {\mathbb A}^r_{k}\) of some algebraic variety \(Z\) and the \(k\)-affine space od dimension \(r\). It is of interest, see the Introduction of the paper under review, to study closed normal projective varieties containing cylinders, and, since the canonical bundle of \(V\) containing a cylinder is not pseudoeffective, total spaces of Mori Fiber Spaces are a class of varieties where examples of such kind can be found. In this context, it sounds natural to study fibrations containing vertical cylinders, i.e, compatible with the fibration structure. In the paper under review the authors start the study of vertical cylinders in a Mori fiber space \(f:X \to Y\) of relative dimension \(3\). There are four classes of Fano \(3\)-folds of Picard number one containing an \({\mathbb A}^3_{\overline{k}}\)-cylinder: the projective space, the quadric, the del Pezzo quintic threefold \(V_5\) of index two and degree five, and a four dimensional family of prime Fano threefolds \(V_{22}\) of genus twelve. To contain a vertical cylinder is equivalent to the fact that the fiber over the generic point of the target \(Y\) contains a \({\mathbb A}^r_K\) cylinder, where \(K\) is the function field of \(Y\). The main result of the paper states that if \(V\) is a \(K\)-form of \(V_5\), that is a smooth projective variety defined over \(K\) whose base extensions to an algebraic closure \(\overline{K}\) is isomorphic (over \(\overline{K}\)) to \(V_5\), then \(V\) always contains an \({\mathbb A}^2_K\)-cylinder, and it is characterized when it contains a \(3\)-cylinder. As a consequence (over \(\overline{k}\)) when \(V_5\) is the general closed fiber of a Mori Fiber Space \(f:X \to C\) over a curve, then \(X\) contains a vertical \({\mathbb A}^3_{\overline{k}}\)-cylinder with respect to \(f\).