  Unconventional superconductivity has recently been discovered in the first
iron-based superconducting silicide LaFeSiH. By using the complementary
techniques of muon spin rotation, tunneling diode oscillator and density
functional theory, we investigate the magnetic penetration depth and thereby
the superconducting gap of this novel high-temperature superconductor. We find
that the magnetic penetration depth displays a sub-$T^2$ behavior in the
low-temperature regime below $T_c/3$, which evidences a nodal structure of the
gap (or a gap with very deep minima). Even if the topology of the computed
Fermi surface is compatible with the $s_\pm$-wave case with accidental nodes,
its nesting and orbital-content features may eventually result in a $d$-wave
state, more unusual for high-temperature superconductors of this class.
