Xenon is a widely used detector target material due to its excellent scintillation properties in the ultraviolet (UV) spectrum. The additional use of infrared (IR) scintillation light could improve future detectors. However, a comprehensive characterization of the IR component is necessary to explore its potential. We report on the first measurement of the time profile of the IR scintillation response of gaseous xenon. Our setup consists of a gaseous xenon target irradiated by an alpha particle source and is instrumented with one IR- and two UV-sensitive photomultiplier tubes. Thereby, it enables IR timing measurements with nanosecond resolution and simultaneous measurement of UV and IR signals. We find that the IR light yield is in the same order of magnitude as the UV yield. We observe that the IR pulses can be described by a fast and a slow component and demonstrate that the size of the slow component decreases with increasing levels of impurities in the gas. Moreover, we study the IR emission as a function of pressure. These findings confirm earlier observations and advance our understanding of the IR scintillation response of gaseous xenon, which could have implications for the development of novel xenon-based detectors.