I am not sure that I understand you correctly, but it seems that your question is answered by the following result (you can find it as Corollary 3.5 in Y.Benyamini, J.Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000): Let $E$ be an infinite-dimensional Banach space. Then there is a Lipschitz retraction of from the unit ball of $E$ to the unit sphere of $E$.

We use this result for $E^0\oplus E^+$. Let $R:B\to\partial B$ be such retraction. Let $T_t:B\to B$ be defined by $T_t(x)=(1-t)x+tRx$. The image $B'_t=T_t(B)$ is homotopic to $B$. The Lipschitz condition implies that for $t$ close to $1$, the image does not contain $0\oplus 1\oplus 0$, which is the only point of $A$ in the subspace.

I am not sure that I understand you correctly, but it seems that your question is answered by the following result (you can find it as Corollary 3.5 in Y.Benyamini, J.Lindenstrauss, Geometric nonlinear functional analysis. Vol. 1. American Mathematical Society Colloquium Publications, 48. American Mathematical Society, Providence, RI, 2000): Let $E$ be an infinite-dimensional Banach space. Then there is a Lipschitz retraction of from the unit ball of $E$ to the unit sphere of $E$.

We use this result for $E^0\oplus E^+$. Let $R:B\to\partial B$ be such retraction. Let $T_t:B\to B$ be defined by $T_t(x)=(1-t)x+tRx$. The image $B'_t=T_t(B)$ is homotopic to $B$. The Lipschitz condition implies that for $t$ close to $1$, the image does not contain $0\oplus 1\oplus 0$, which is the only candidate for the intersection of $A$ and $B'_t$.