Examples of isometric shifts on C(X)

We give two independent methods for obtaining examples of separable spaces X for which C(X) admits an isometric shift. The first one allows us in particular to construct examples with infinitely many nonhomeomorphic components in any infinite-dimensional normed space. The second one applies for instance to sequences adjoined to any n-dimensional compact manifold (for n⩾2) or to the Sierpiński curve. The combination of both techniques leads to examples with different special features involving a convergent sequence adjoined to the Cantor set.