  In the present work we demonstrate that C-doped Zr$_{5}$Pt$_{3}$ is an
electron-phonon superconductor (with critical temperature T$_\mathrm{C}$ =
3.7\,K) with a nonsymmorphic topological Dirac nodal-line semimetal state,
which we report here for the first time. The superconducting properties of
Zr$_{5}$Pt$_{3}$C$_{0.5}$ have been investigated by means of magnetization and
muon spin rotation and relaxation ($\mu$SR) measurements. We find that at low
temperatures the depolarization rate is almost constant and can be well
described by a single-band $s-$wave model with a superconducting gap of
$2\Delta(0)/k_\mathrm{B}T_\mathrm{C}$ = 3.84, close to the value of BCS theory.
From transverse field $\mu$SR analysis we estimate the London penetration depth
$\lambda_{L}$ = 469 nm, superconducting carrier density $n_{s}$ =
2$\times$10$^{26}$ $m^{-3}$, and effective mass m$^{*}$ = 1.584 $m_{e}$. Zero
field $\mu$SR confirms the absence of any spontaneous magnetic moment in the
superconducting ground state. To gain additional insights into the electronic
ground state of C-doped Zr$_5$Pt$_3$, we have also performed first-principles
calculations within the framework of density functional theory (DFT). The
observed homogenous electronic character of the Fermi surface as well as the
mutual decrease of $T_\mathrm{C}$ and density of states at the Fermi level are
consistent with the experimental findings. However, the band structure reveals
the presence of robust, gapless fourfold-degenarate nodal lines protected by
$6_{3}$ screw rotations and glide mirror planes. Therefore, Zr$_5$Pt$_3$
represents a novel, unprecedented condensed matter system to investigate the
intricate interplay between superconductivity and topology.
