We investigate the thermodynamical and magnetic properties of a ``dark-cored" fibril seen in the chromospheric Ca II IR line at 854.2 nm to determine the physical process behind its appearance. We analyse a time series of spectropolarimetric observations obtained in the Ca II IR line at 854.2 nm and the photospheric Fe I line at 630.25 nm. We simultaneously invert the spectra in both wavelength ranges with the SIR code to obtain the temperature and velocity stratification with height in the solar atmosphere and the magnetic field properties in the photosphere. The structure can be clearly traced in the line-of-sight (LOS) velocity and the temperature maps. It connects from a small pore with kG fields to a region with lower field strength. The flow velocity and the temperature indicate that the height of the structure increases with increasing distance from the inner footpoint. The Stokes V signal of 854.2 nm shows a Doppler-shifted polarization signal with the same displacement as in the intensity profile, indicating that the supersonic flow seen in the LOS velocity is located within magnetized plasma. We conclude that the chromospheric dark-cored fibril traces a siphon flow along magnetic field lines, driven by the gas pressure difference caused by the higher magnetic field strength at the inner footpoint. We suggest that fast flows guided by the magnetic field lead to the appearance of ``dark-cored" fibrils in intensity images. Although the observations included the determination of the polarization signal in the chromospheric Ca II IR line, the signal could not be analysed quantitatively due to the low S/N. Chromospheric polarimetry will thus require telescopes of larger aperture able to collect a sufficient number of photons for a reliable determination of polarization in deep and only weakly polarized spectral lines.