Does the canonical $\pi: X \to X/Y$ map the closed unit ball to the closed unit ball?

Question

Let $Y \subset X$ be a closed subspace of the normed space $X$. Consider $\pi: X \to X/Y, x \mapsto [x]$.

Then for $x \in X, ||x||\le 1$: $\quad||[x]|| = \text{inf}_{y \in Y} ||x-y|| \le \text{inf}_{y \in Y} ||x|| + ||y|| \le ||x|| \le 1$

Does $||\pi(x)|| \le 1 \Rightarrow ||x|| \le 1$ hold aswell, i.e. $\pi(\bar B_{1}(0)) = \bar B_{1}(0)$?