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$^{23}$Na spin-1 Bose-Einstein condensates are used to experimentally demonstrate that mean-field physics beyond the single-mode approximation can be relevant during the non-equilibrium dynamics. The experimentally observed spin oscillation dynamics and associated dynamical spatial structure formation confirm theoretical predictions that are derived by solving a set of coupled mean-field Gross-Pitaevskii equations [J. Jie et al., Phys. Rev. A 102, 023324 (2020)]. The experiments rely on microwave dressing of the $f=1$ hyperfine states, where $f$ denotes the total angular momentum of the $^{23}$Na atom. The fact that beyond single-mode approximation physics at the mean-field level, i.e., spatial mean-field dynamics that distinguishes the spatial density profiles associated with different Zeeman levels, can -- in certain parameter regimes -- have a pronounced effect on the dynamics when the spin healing length is comparable to or larger than the size of the Bose-Einstein condensate has implications for using Bose-Einstein condensates as models for quantum phase transitions and spin squeezing studies as well as for non-linear SU(1,1) interferometers., Comment: 9 pages; 8 figures, several subfigures