  The superconducting properties of LaFeAsO$_{1-x}$F$_{x}$ in conditions of
optimal electron-doping are investigated upon the application of external
pressure up to $\sim 23$ kbar. Measurements of muon-spin spectroscopy and dc
magnetometry evidence a clear mutual independence between the critical
temperature $T_{c}$ and the low-temperature saturation value for the ratio
$n_{s}/m^{*}$ (superfluid density over effective band mass of Cooper pairs).
Remarkably, a dramatic increase of $\sim 30$ % is reported for $n_{s}/m^{*}$ at
the maximum pressure value while $T_{c}$ is substantially unaffected in the
whole accessed experimental window. We argue and demonstrate that the
explanation for the observed results must take the effect of non-magnetic
impurities on multi-band superconductivity into account. In particular, the
unique possibility to modify the ratio between intra-band and inter-bands
scattering rates by acting on structural parameters while keeping the amount of
chemical disorder constant is a striking result of our proposed model.
