Recent theoretical studies proposed that two-dimensional (2D) GaGeTe crystals have promising high detection sensitivity at infrared wavelengths and can offer ultra-fast operation. This can be attributed to their small optical bandgap and high carrier mobility. However, experimental studies on GaGeTe in the infrared region are lacking and this exciting property has not been explored yet. In this work, we demonstrate a short-wavelength infrared (SWIR) photodetector based on a multilayer (ML) GaGeTe field-effect transistor (FET). Fabricated devices show a p-type behavior at room temperature with a hole field-effect mobility of 8.6 - 20 cm2 V-1s-1. Notably, under 1310 nm illumination, the photo responsivities and noise equivalent power of the detectors with 65 nm flake thickness can reach up to 57 A/W and 0.1 nW/Hz1/2, respectively, at a drain-source bias (Vds) = 2 V. The frequency responses of the photodetectors were also measured with a 1310 nm intensity-modulated light. Devices exhibit a response up to 100 MHz with a 3dB cut-off frequency of 0.9 MHz. Furthermore, we also tested the dependence of the device frequency response on the applied bias and gate voltages. These early experimental findings stimulate the potential use of multilayer GaGeTe for highly sensitive and ultrafast photodetection applications.