We report the first detection of polarization signals induced by the Zeeman Effect in lines of the E4Π-A4Π system of FeH located at 1.6 μm. Motivated by the tentative detection of this band in the intensity spectrum of late-type dwarfs, we have investigated in detail the full Stokes sunspot spectrum in the near-infrared around 1.6 μm, finding conspicuous signals which we associate with the FeH lines of the E4Π-A4Π band. We investigate the Zeeman Effect in these molecular transitions in order to identify the observed lines and to point out their potential interest for magnetic field diagnostics. First, we use nearby OH lines for estimating the magnetic field strength in the sunspot umbra. We then apply the weak-field approximation to obtain a first estimation of the effective Landé factor of some spectral lines, which we identify as FeH lines. Our tentative identification is performed under the assumption that Hund's case coupling is a sufficiently good approximation. Theoretical and/or laboratory investigations of this molecular system are urgently needed for a full exploitation of the diagnostic capabilities of the band.