We present an optical nebular spectrum of the nearby Type Ia supernova 2011fe, obtained 981 d after explosion. SN 2011fe exhibits little evolution since the +593 d optical spectrum, but there are several curious aspects in this new extremely late-time regime. We suggest that the persistence of the ~5800 Å feature is due to Na I D, and that a new emission feature at ~7300 Å may be [Ca II]. Also, we discuss whether the new emission feature at ~6400 Å might be [Fe I] or the high-velocity hydrogen predicted by Mazzali et al. The nebular feature at 5200 Å exhibits a linear velocity evolution of ~350 km s-1 per 100 d from at least +220 to +980 d, but the line's shape also changes in this time, suggesting that line blending contributes to the evolution. At ~1000 d after explosion, flux from the SN has declined to a point where contribution from a luminous secondary could be detected. In this work, we make the first observational tests for a post-impact remnant star and constrain its temperature and luminosity to T ≳ 104 K and L ≲ 104 L⊙. Additionally, we do not see any evidence for narrow H α emission in our spectrum. We conclude that observations continue to strongly exclude many single-degenerate scenarios for SN 2011fe.