  The coherence length $\xi$ is the fundamental length scale of superconductors
which governs the sizes of Cooper pairs, vortices, Andreev bound states, and
more. In existing microscopic theories of superconductivity, it is expected
that as the strength of the attractive interaction increases, $\xi$ decreases
as the electrons are bound together more strongly. In BCS theory, the coherence
length is $\xi_\mathrm{BCS} = \hbar v_{F}/\Delta$, where $v_{F}$ is the Fermi
velocity and $\Delta$ is the pairing gap. It is clear that increasing $\Delta$
will shorten $\xi_\mathrm{BCS}$. However, the situation is puzzling for
superconductors with exactly flat bands in which $v_{F}$ goes to zero and
$\xi_\mathrm{BCS}$ is expected to be zero. In this work, we show that the
quantum metric, which is the real part of the quantum geometric tensor, gives
rise to an anomalous contribution to the coherence length. Specifically, $\xi =
\sqrt{\xi_\mathrm{BCS}^2 +\ell_{\mathrm{qm}}^{2}}$ for a superconductor where
$\ell_{\mathrm{qm}}$ is the quantum metric contribution. In the flat-band
limit, $\xi$ does not vanish but is bound below by $\ell_{\mathrm{qm}}$. We
demonstrate that under the uniform pairing condition, $\ell_{\mathrm{qm}}$ is
controlled by the quantum metric of minimal trace in the flat-band limit.
Physically, the Cooper pair size of a superconductor cannot be squeezed down to
a size smaller than $\ell_{\mathrm{qm}}$ which is a fundamental length scale
determined by the quantum geometry of the wave functions. Lastly, we compute
the quantum metric contributions for the family of superconducting moir\'{e}
graphene materials, demonstrating the significant role played by quantum metric
effects in these narrow-band superconductors.
