Time-dependent analytic solutions of the Einstein-Skyrme system -gravitating Skyrmions-, with topological charge one are analyzed in detail. In particular, the question of whether these Skyrmions reach a spherically symmetric configuration for t → +∞ is discussed. It is shown that there is a static, spherically symmetric solution described by the Ermakov-Pinney system, which is fully integrable by algebraic methods. For Λ > 0 this spherically symmetric solution is found to be in a "neutral equilibrium" under small deformations, in the sense that under a small squashing it would neither blow up nor dissapear after a long time, but it would remain finite forever (plastic deformation). Thus, in a sense, the coupling with Einstein gravity spontaneously breaks the spherical symmetry of the solution. However, in spite of the lack of isotropy, for t → ∞ (and Λ > 0) the space time is locally flat and the anisotropy of the Skyrmion only reflects the squashing of spacetime.