Context. Stray light caused by scattering on optical surfaces and in the Earth's atmosphere degrades the spatial resolution of observations. Whereas post-facto reconstruction techniques are common for 2D imaging and spectroscopy, similar options for slit-spectrograph data are rarely applied. Aims: We study the contribution of stray light to the two channels of the POlarimetric LIttrow Spectrograph (POLIS) at 396 nm and 630 nm as an example of a slit-spectrograph instrument. We test the performance of different methods of stray-light correction and spatial deconvolution to improve the spatial resolution post-facto. Methods: We model the stray light as having two components: a spectrally dispersed component and a "parasitic" component of spectrally undispersed light caused by scattering inside the spectrograph. We used several measurements to estimate the two contributions: a) observations with a (partly) blocked field of view (FOV); b) a convolution of the FTS spectral atlas; c) imaging of the spider mounting in the pupil plane; d) umbral profiles; and e) spurious polarization signal in telluric spectral lines. The measurements with a partly blocked FOV in the focal plane allowed us to estimate the spatial point spread function (PSF) of POLIS and the main spectrograph of the German Vacuum Tower Telescope (VTT). We then used the obtained PSF for a deconvolution of both spectroscopic and spectropolarimetric data and investigated the effect on the spectra. Results: The parasitic contribution can be directly and accurately determined for POLIS, amounting to about 5% (0.3%) of the (continuum) intensity at 396 nm (630 nm). The spectrally dispersed stray light is less accessible because of its many contributing sources. We estimate a lower limit of about 10% across the full FOV for the dispersed stray light from umbral profiles. In quiet Sun regions, the stray-light level from the close surroundings (d < 2'') of a given spatial point is about 20%. The stray light reduces to below 2% at a distance of 20'' from a lit area for both POLIS and the main spectrograph. The spatial deconvolution using the PSF obtained improves the spatial resolution and increases the contrast, with a minor amplification of noise. Conclusions: A two-component model of the stray-light contributions seems to be sufficient for a basic correction of observed spectra. The instrumental PSF obtained can be used to model the off-limb stray light, to determine the stray-light contamination accurately for observation targets with large spatial intensity gradients such as sunspots, and also to improve the spatial resolution of observations post-facto.