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We consider 1-complemented subspaces (ranges of contractive projections) ofvector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ istwice continuously differentiable and satisfies certain conditions connectingthe norm and the notion of disjointness with respect to the basis, then weprove that every 1-complemented subspace of $\ell_p(X)$ admits a basis ofmutually disjoint elements. Moreover, we show that every contractive projectionis then an averaging operator. We apply our results to the space$\ell_p(\ell_q)$ with $p,q\in (1,2)\cup (2,\infty)$ and obtain a completecharacterization of its 1-complemented subspaces.

Second derivatives of norms and contractive complementation in vector-valued spaces