We present spin-noise spectroscopy measurements on an ensemble of donor-bound electrons in ultrapure GaAs:Si covering temporal dynamics over 6 orders of magnitude from milliseconds to nanoseconds. The spin-noise spectra detected at the donor-bound exciton transition show the multifaceted dynamical regime of the ubiquitous mutual electron and nuclear spin interaction typical for III-V-based semiconductor systems. The experiment distinctly reveals the finite Overhauser shift of an electron spin precession at zero external magnetic field and a second contribution around zero frequency stemming from the electron spin components parallel to the nuclear spin fluctuations. Moreover, at very low frequencies, features related with time-dependent nuclear spin fluctuations are clearly resolved making it possible to study the intricate nuclear spin dynamics at zero and low magnetic fields. The findings are in agreement with the developed model of electron and nuclear spin noise. DOI: 10.1103/PhysRevLett.115.176601 PACS numbers: 72.25.Rb, 72.70.+m, 78.47.db, 85.75.-d Harnessing coherence is one of the central topics in current research and attracts high interest due to the complex fundamental physics bridging quantum mechanics and statistics as well as due to prospective applications for information processing [1][2][3]. The solid state quantum states based upon the spin degree of freedom of confined carriers in semiconductors are at the forefront of many current research activities in this field. In this respect, optically addressable electron and hole spin quantum states in III-V-based semiconductor systems bear the beauty of efficient options for initialization, manipulation, and readout by light in combination with exceptional sample quality [4]. Currently, a promising system for these tasks are donorbound electrons in ultrahigh quality, very weakly n-doped GaAs since the widely spaced, quasi-isolated electrons act as an ensemble of identical, individually localized atoms [5,6]. However, the ostensible catch of this approach is the inherent interaction with the nuclear spin bath which has been addressed in many different systems so far [7][8][9][10][11].In principle, there are different approaches to deal with the decoherence imposed via the hyperfine interaction. On the first sight, the most obvious way is to replace the isotopes carrying a nuclear spin with spinless isotopes like in 28 Si [12] but silicon has the drawback of an indirect gap. Direct semiconductors with spinless isotopes like, e.g., isotopically purified II-VI systems have yet the drawback of inferior sample quality. In single III-V-based quantum dots, the hyperfine interaction can be reduced by either moving on to hole spins which show a diminished hyperfine interaction [13][14][15] or by polarizing the nuclei in order to make them less effective [16,17]. Besides that, the mutual interaction between carrier and nuclear spins is also strain dependent and strongly varying coupling strengths in such nanostructures result in a row of widely discussed pr...