  Benyamini and Lindenstrauss mention in their monograph \emph{Geometric
nonlinear functional analysis Vol. 1., American Mathematical Society Colloquium
Publications, 48. American Mathematical Society, Providence, RI, 2000} 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.
