  We initiate the large scale geometric study of Banach-Lie groups, especially
of linear Banach-Lie groups. We show that the exponential length, originally
introduced by Ringrose for unitary groups of $C^*$-algebras, defines the
quasi-isometry type of any connected Banach-Lie group. As an illustrative
example, we consider unitary groups of separable abelian unital $C^*$-algebras
with spectrum having finitely many components, which we classify up to
topological isomorphism and up to quasi-isometry, in order to highlight the
difference. The main results then concern the Haagerup property, and Properties
(T) and (FH). We present the first non-trivial non-abelian and non-localy
compact groups having the Haagerup property, most of them being non-amenable.
These are the groups $\mathcal{U}_2(M,\tau)$, where $M$ is a semifinite von
Neumann algebra with a normal faithful semifinite trace $\tau$. Finally, we
investigate the groups $\mathrm{E}_n(A)$, which are closed subgroups of
$\mathrm{GL}(n,A)$ generated by elementary matrices, where $A$ is a unital
Banach algebra. We show that for $n\geq 3$, all these groups have Property (T)
and they are unbounded, so they have Property (FH) non-trivially. On the other
hand, if $A$ is an infinite-dimensional unital $C^*$-algebra, then
$\mathrm{E}_2(A)$ does not have the Haagerup property. If $A$ is moreover
abelian and separable, then $\mathrm{SL}(2,A)$ does not have the Haagerup
property.
