The availability of high-frequency pulsed emitters in the 2-2.5 µm wavelength range paved the way for a wealth of new applications in ultrafast spectroscopy, free-space and fiber-optical communications, surveillance and recognition, artificial intelligence, and medical imaging. However, developing these emerging technologies and their large-scale use depend on the availability of high-speed, low-noise, and cost-effective photodetectors. With this perspective, here we demonstrate GeSn photodiodes grown on silicon wafers featuring a high broadband operation covering the extended-SWIR range with a peak responsivity of 0.3 A/W at room temperature. These GeSn devices exhibit a high bandwidth reaching 7.5 GHz at 5 V bias with a 2.6 µm cutoff wavelength, and their integration in ultrafast time-resolved spectroscopy applications is demonstrated. In addition to enabling time-resolved electro-luminescence at 2.3 µm, the high-speed operation of GeSn detectors was also exploited in the diagnostics of ultra-short pulses of a supercontinuum laser with a temporal resolution in the picosecond range at 2.5 µm. Establishing these capabilities highlights the potential of manufacturable GeSn photodiodes for silicon-integrated high-speed extended-SWIR applications.