Isometric embedding into spaces of continuous functions

We prove that some Banach spaces X have the property that every Banach space that can be isometrically embedded in X can be isometrically and linearly embedded in X. We do not know if this is a general property of Banach spaces. As a consequence we characterize for which ordinal numbers α, β there exists an isometric embedding between ${\displaystyle C_0(\alpha +1)}$ and ${\displaystyle C_0(\beta +1)}$.