REMS-P, the pressure measurement subsystem of the Mars Science Laboratory (MSL) Rover Environmental Measurement Station (REMS), is performing accurate observations of the Martian atmospheric surface pressure. It has demonstrated high data quality and good temporal coverage, carrying out the first in situ pressure observations in the Martian equatorial regions. We describe the REMS-P initial results by MSL mission sol 100 including the instrument performance and data quality and illustrate some initial interpretations of the observed features. The observations show both expected and new phenomena at various spatial and temporal scales, e.g., the gradually increasing pressure due to the advancing Martian season signals from the diurnal tides as well as various local atmospheric phenomena and thermal vortices. Among the unexpected new phenomena discovered in the pressure data are a small regular pressure drop at every sol and pressure oscillations occurring in the early evening. We look forward to continued high-quality observations by REMS-P, extending the data set to reveal characteristics of seasonal variations and improved insights into regional and local phenomena.