We present a near-infrared spectral sequence of the electromagnetic counterpart to the binary neutron star merger GW170817 detected by Advanced LIGO/Virgo. Our dataset comprises seven epochs of J+H spectra taken with FLAMINGOS-2 on Gemini-South between 1.5 and 10.5 days after the merger. In the initial epoch, the spectrum is dominated by a smooth blue continuum due to a high-velocity, lanthanide-poor blue kilonova component. Starting the following night, all subsequent spectra instead show features that are similar to those predicted in model spectra of material with a high concentration of lanthanides, including spectral peaks near 1.07 and 1.55 microns. Our fiducial model with 0.04 M_sun of ejecta, an ejection velocity of v=0.1c, and a lanthanide concentration of X_lan=1e-2 provides a good match to the spectra taken in the first five days, although it over-predicts the late-time fluxes. We also explore models with multiple fitting components, in each case finding that a significant abundance of lanthanide elements is necessary to match the broad spectral peaks that we observe starting at 2.5 d after the merger. These data provide direct evidence that binary neutron star mergers are significant production sites of even the heaviest r-process elements.