We have discovered spontaneous ordering of nuclear spins in lithium metal by NMR measurements at very low temperatures. In low magnetic fields, B<0.2 mT, the NMR spectra show a pronounced low-frequency anomaly. Also, nonadiabatic response to a slowly varying magnetic field was observed. A rich phase diagram with three different nonparamagnetic regions is proposed. We estimate a critical spin temperature T(c) approximately 350 nK at B=0. We also report the absence of superconductivity in lithium at normal pressure down to T(e) approximately 100 microK (B<10 nT).