There is a useful simple
Lemma. If $X\sim X\oplus X$, $Y\sim Y\oplus Y$, each of $X,Y$ is isomorphic to a complemented subspace of another.
Proof. We have $X\sim Y\oplus A$, then $X\sim (Y\oplus Y)\oplus A=Y\oplus(Y\oplus A)=Y\oplus X$, analogously $Y\sim X\oplus Y$.
Now let $X=C([0,1])$, $Y=c_0(X)$. Property $Y\sim Y\oplus Y$ is clear, property $X\sim X\oplus X$ follows from $X\oplus X=C([0,1]\times \{0,1\})$ and Milyutin theorem. $X$ is clearly complemented in $Y$. At last, $Y$ is the space of functions on the compact space on the plane $K=\cup \{\frac1n \times [0,\frac1n]\}\cup (0,0)$, which are equal to 0 at $(0,0)$. Again by Milyutin theorem we see that $X\sim C(K)$, hence $Y$ is isomorphic to a hyperplane in the space isomorphic to $X$. Hyperplane is of course complemented.
This is maybe not a very good proof, since your claim may be used in the proof of Milyutin theorem (I do not remember). But if so, study the proof, it should contain this claim.