We present the first interferometric NIR observations of the LEV η Carinae with high spectral resolution. The observations were carried out with three 8.2 m VLTI Unit Telescopes in the K-band. The raw data are spectrally dispersed interferograms obtained with spectral resolutions of 1,500 (MR-K mode) and 12,000 (HR-K mode). The observations were performed in the wavelength range around both the He I 2.059 μm and the Brγ 2.166 μm emission lines. The spectrally dispersed AMBER interferograms allow the investigation of the wavelength dependence of the visibility, differential phase, and closure phase of η Car. In the K-band continuum, a diameter of 4.0 ± 0.2 mas (Gaussian FWHM) was measured for η Car's optically thick wind region, whereas the Brγ and He I emission line regions are larger. If we fit Hillier et al. model visibilities to the observed AMBER visibilities, we obtain 50% encircled-energy diameters of 4.3, 6.5 and 9.6 mas in the 2.17 μm continuum, the He I, and the Brγ emission lines, respectively. In the continuum near the Brγ line, an elongation along a position angle of 128° ± 15° was found, consistent with previous VLTI/VINCI measurements. We find good agreement between the measured visibilities and the predictions of the radiative transfer model of Hillier et al. For the interpretation of the non-zero differential and closure phases measured within the Brγ line, we present a simple geometric model of an inclined, latitude-dependent wind zone. Our observations support theoretical models of anisotropic winds from fast-rotating, luminous hot stars with enhanced high-velocity mass loss near the polar regions.