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The Close AGN Reference Survey (CARS): Locating the [O III] wing component in luminous local Type 1 AGN

[Abridged]The strong asymmetry in the optical [O III]$λ$5007 emission line is one of the best signatures of AGN-driven warm (~10$^4$ K) ionized gas outflows on host galaxy scales. While large spectroscopic surveys like SDSS have characterized the kinematics of [O III] for large samples of AGN, estimating the associated energetics requires spatially resolving these outflows with, for example, IFU studies. As part of CARS we obtained spatially-resolved IFU spectroscopy for a representative sample of 39 luminous type 1 AGN at 0.01<z<0.06 with MUSE and VIMOS IFUs at the VLT to infer the spatial location of the ionized gas outflows. We compare the light distributions of the [O III] wing to that of the H$β$ broad emission line region, a classical point source (PSF). We then use the PSF to distinguish between the unresolved and resolved [O III] wing emission. We further determine its location using spectro-astrometry for the point-like sources. The [O III] wing is spatially unresolved in 23 out of the 36 AGN with >80 % of the flux associated with a point-like source. We measure <100 pc offsets in the spatial location of the outflow from the AGN nucleus using the spectro-astrometry technique for these sources. For the other 13 AGN, the [O III] wing emission is resolved and possibly extended on kpc scale. We conclude that [O III] wing emission can be compact or extended in an unbiased luminous AGN sample, where both cases are likely to appear. Electron density in the compact [O III] wing regions (median $n_e$~1900 cm$^{-3}$) is nearly a magnitude higher than in the extended ones (median $n_e$~500 cm$^{-3}$). The presence of spatially extended and compact [O III] wing emission is unrelated to the AGN bolometric luminosity and to inclination effects, which means other features such as time delays, or mechanical feedback/radio jets may shape the ionized gas outflow properties.