Unconventional superconductivity has recently been discovered in the 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 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(+/-)-wave case with accidental nodes, its nesting and orbital-content features may eventually result in a d-wave state, which is more unusual for high-temperature superconductors of this class.