We present results of low-temperature scanning tunneling microscopy/spectroscopy measurements on the one-atom-layer superconductor, Si(111)-root 3 x root 3-(Tl, Pb) which has a spin-split band structure due to the Rashba-Bychkov effect. It was revealed that it has a multiple superconducting gap Delta with significant anisotropy and sizable magnitude of 2 Delta/k(B)T(C) similar to 8.6 (k(B) is the Boltzmann constant and T-C is the critical superconducting temperature). Under the magnetic field, a dip structure was observed even at the core of the vortex. The dip structure was like a ``pseudogap{''} because it remained up to 2 T, well above the upper critical magnetic field determined in the transport measurements (similar to 0.7 T). This unconventional behavior suggests the possibility of spin-triplet Cooper pairs related to parity-broken superconductors.