The shortwave infrared (SWIR) spectrometer module of the Tropospheric Monitoring Instrument (TROPOMI), on board the ESA Copernicus Sentinel-5 Precursor satellite, is used to measure atmospheric CO and methane columns. For this purpose, calibrated radiance measurements are needed that are minimally contaminated by instrumental stray light. Therefore, a method has been developed and applied in an on-ground calibration campaign to characterize stray light in detail using a monochromatic quasi-point light source. The dynamic range of the signal was extended to more than 7 orders of magnitude by performing measurements with different exposure times, saturating detector pixels at the longer exposure times. Analysis of the stray light indicates about 4.4 % of the detected light is correctable stray light. An algorithm was then devised and implemented in the operational data processor to correct in-flight SWIR observations in near-real time, based on Van Cittert deconvolution. The stray light is approximated by a far-field kernel independent of position and wavelength and an additional kernel representing the main reflection. Applying this correction significantly reduces the stray-light signal, for example in a simulated dark forest scene close to bright clouds by a factor of about 10. Simulations indicate that this reduces the stray-light error sufficiently for accurate gas-column retrievals. In addition, the instrument contains five SWIR diode lasers that enable long-term, in-flight monitoring of the stray-light distribution.