Let $G$ be a connected Banach-Lie group with a Banach-Lie algebra $\mathfrak{g}$ (infinite-dimensional in general), which is in particular a Banach space with some compatible norm $\|\cdot\|$. Assuming `sufficient smoothness', one can define the length of curves in $G$, using the norm $\|\cdot\|$ on $\mathfrak{g}$, in essentially the same way as one defines the length of curves on finite dimensional connected Lie groups using an inner product (or rather the induced Hilbert norm) on $\mathfrak{g}$. Then one can define a left-invariant distance function on $G$ as the infimum of lengths of curves joining two points.
In the finite-dimensional setting, this always gives a genuine metric which is compatible with the topology of $G$. In the infinite-dimensional setting, it is not clear whether this is distance is non-degenerate (positive on pairs of different elements), or if it induces the original topology of $G$.
This must have been thoroughly studied, but I couldn't find any definitive answer (only could find some hints that it might not be true for some Frechet-Lie groups). Can something be said about all Banach-Lie groups?