An example of a non-commutative uniform Banach group

Benyamini and Lindenstrauss mention in their monograph (Benyamini and Lindenstrauss, 2000 [2]) that there is no known example of a non-commutative uniform Banach group. Prassidis and Weston also asked whether there is a non-commutative example. We answer this problem affirmatively. We construct a non-commutative uniform Banach group which has the free group of countably many generators as a dense subgroup. Moreover, we show that our example is a free one-generated uniform Banach group whose metric induced by the norm is bi-invariant.