Polarization-dependent near-edge x-ray absorption spectroscopy measurements have been performed on the O 1s and Ni 2p edges of ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{NiO}}_{4+\mathrm{\ensuremath{\delta}}}$ single crystals for 0\ensuremath{\le}x\ensuremath{\le}0.6. The results are compared with recent data on ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{CuO}}_{4+\mathrm{\ensuremath{\delta}}}$, and NiO and with detailed cluster calculations on a ${\mathrm{NiO}}_{6}$ cluster. From this, we determine the energetic ordering of the states close to the Fermi level with Ni(Cu) 3${\mathit{d}}_{{\mathit{x}}^{2}\mathrm{\ensuremath{-}}{\mathit{y}}^{2}}$/O 2${\mathit{p}}_{\mathit{x},\mathit{y}}$ and Ni(Cu) 3${\mathit{d}}_{3{\mathit{z}}^{2}\mathrm{\ensuremath{-}}{\mathit{r}}^{2}}$/apex-O 2${\mathit{p}}_{\mathit{z}}$ orbital character for the undoped compounds. We find that d-d exciton side bands show up in the final states of ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{NiO}}_{4+\mathrm{\ensuremath{\delta}}}$, giving evidence for the Zhang-Rice character of the doped carriers. The spectral-weight transfer in the O 1s spectra shows a significant dynamical component. Thus, we suggest that the polarons in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathit{x}}$${\mathrm{Sr}}_{\mathit{x}}$${\mathrm{NiO}}_{4+\mathrm{\ensuremath{\delta}}}$ can be seen as nonclassical objects on the length scale of a lattice constant and that the formation of polaron domain walls and polaron lattices possibly has to be considered as a result of this large coherence length of the polarons. \textcopyright{} 1996 The American Physical Society.