  By performing P doping on the Si sites in the topological semimetal
Mo$_5$Si$_3$, we discover strong-coupling superconductivity in
Mo$_5$Si$_{3-x}$P$_x$ (0.5 $\le$ $x$ $\le$ 2.0). Mo$_5$Si$_3$ crystallizes in
the W$_5$Si$_3$-type structure with space group of $I4/mcm$ (No. 140), and is
not a superconductor itself. Upon P doping, the lattice parameter $a$ decreases
while $c$ increases monotonously. Bulk superconductivity is revealed in
Mo$_5$Si$_{3-x}$P$_x$ (0.5 $\le$ $x$ $\le$ 2.0) from resistivity,
magnetization, and heat capacity measurements. $T_c$ in
Mo$_5$Si$_{1.5}$P$_{1.5}$ reaches as high as 10.8 K, setting a new record among
the W$_5$Si$_3$-type superconductors. The upper and lower critical fields for
Mo$_5$Si$_{1.5}$P$_{1.5}$ are 14.56 T and 105 mT, respectively. Moreover,
Mo$_5$Si$_{1.5}$P$_{1.5}$ is found to be a fully gapped superconductor with
strong electron-phonon coupling. First-principles calculations suggest that the
enhancement of electron-phonon coupling is possibly due to the shift of the
Fermi level, which is induced by electron doping. The calculations also reveal
the nontrivial band topology in Mo$_5$Si$_3$. The $T_c$ and upper critical
field in Mo$_5$Si$_{3-x}$P$_x$ are fairly high among pseudobinary compounds.
Both of them are higher than those in NbTi, making future applications
promising. Our results suggest that the W$_5$Si$_3$-type compounds are ideal
platforms to search for new superconductors. By examinations of their band
topologies, more candidates for topological superconductors can be expected in
this structural family.
