We present results of low-temperature scanning tunneling microscopy/spectroscopy measurements on the one-atom-layer superconductor, Si (111 )-√ 3 ×√ 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 ∆ with significant anisotropy and sizable magnitude of 2 ∆ /kBTC∼8.6 (kB is the Boltzmann constant and TC 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 (∼0.7 T). This unconventional behavior suggests the possibility of spin-triplet Cooper pairs related to parity-broken superconductors.