  Nodal-line semimetals are topologically non-trivial states of matter
featuring band crossings along a closed curve, i.e. nodal-line, in momentum
space. Through a detailed analysis of the electronic structure, we show for the
first time that the normal state of the superconductor NaAlSi, with a critical
temperature of $T_{\rm c} \approx$ 7 K, is a nodal-line semimetal, where the
complex nodal-line structure is protected by non-symmorphic mirror crystal
symmetries. We further report on muon spin rotation experiments revealing that
the superconductivity in NaAlSi is truly of bulk nature, featuring a fully
gapped Fermi-surface. The temperature-dependent magnetic penetration depth can
be well described by a two-gap model consisting of two $s$-wave symmetric gaps
with $\Delta_1 =$ 0.6(2) meV and $\Delta_2 =$ 1.39(1) meV. The zero-field muon
experiment indicates that time-reversal symmetry is preserved in the
superconducting state. Our observations suggest that notwithstanding its
topologically non-trivial band structure, NaAlSi may be suitably interpreted as
a conventional London superconductor, while more exotic superconducting gap
symmetries cannot be excluded. The intertwining of topological electronic
states and superconductivity renders NaAlSi a prototypical platform to search
for unprecedented topological quantum phases.
