Simply connected closed subspace of Banach space is a retract

Question

Suppose that $K$ is a simply connected and closed subset of an infinite-dimensional Banach space $B$. Then is $K$ necessarily a retract of $B$?

I can't seem to find a counter example..

Answer 1

1. Each Banach space is contractible.

2. Each retract of a contractible space is contractible.

3. Each Banach space of dimension $> 1$ contains a compact simply connected subset which is not contractible.