A density problem

Question

Let $\langle\cdot,\cdot\rangle$ be the usual scalar product in ${\bf R}^n$ ($n\geq 2$) and let $B$ be the closed unit ball of ${\bf R}^n$. Denote by $C^0(B,B)$ the space of all continuous functions from $B$ into $B$ endowed with the usual sup-metric. Denote by $\Lambda$ the set of all $f\in C^0(B,B)$ for which there exists some continuous function $\alpha_f:B\to {\bf R}$ such that the set $$\{(x,y)\in B\times {\bf R}^n : \langle f(x)-x,y\rangle=\alpha_f(x)\}$$ is disconnected. Is it true that $\Lambda$ is dense in $C^0(B,B)$ ?