  The first-principles calculations and measurements of the magnetic
penetration depths, the upper critical field, and the specific heat were
performed for a family of Mo$_5$Si$_{3-x}$P$_x$ superconducotrs.
First-principles calculations suggest the presence of a flat band dispersion,
which gradually shifts to the Fermi level as a function of phosphorus doping
$x$. The flat band approaches the Fermi level at $x\simeq 1.3$, thus separating
Mo$_5$Si$_{3-x}$P$_x$ between the purely steep band and the steep band/flat
band superconducting regimes. The emergence of flat bands lead to an abrupt
change of nearly all the superconducting quantities. In particular, a strong
reduction of the coherence length $\xi$ and enhancement of the penetration
depth $\lambda$ result in nearly factor of three increase of the
Ginzburg-Landau parameter $\kappa=\lambda/\xi$ (from $\kappa\simeq 25$ for
$x\lesssim 1.2$ to $\kappa\simeq 70$ for $x\gtrsim 1.4$) thus initiating the
transition of Mo$_5$Si$_{3-x}$P$_x$ from a moderate to an extreme type-II
superconductivity.
