Massive early-type galaxies are believed to be the end result of an extended mass accretion history. The stars formed in situ very early on in the initial phase of the assembly might have originated from an extremely intense star formation burst, and may still be found within the cores of such galaxies today. We investigate the presence of a surviving high-$z$ compact progenitor component in the brightest galaxy of the Hydra I cluster, NGC 3311, by mapping its 2D kinematics and stellar population out to 2 effective radii, combining MUSE observations, extended EMILES models, and a newly developed parametric fully Bayesian framework using full-spectrum fitting. We present 2D maps and radial profiles of the stellar velocity dispersion, age, total metallicity, $alpha$-element, sodium abundance ([Na/Fe]), and the initial mass function (IMF) slope. All properties have significant gradients, confirming the existence of multiple structural components, including a young, metal-rich blue spot. We find that the component dominating the light budget of NGC 3311 within $Rlesssim 2.0$ kpc is the surviving $z=0$ analog of a high-$z$ compact core. This concentrated structure has a relatively small velocity dispersion ($sigma_*approx 180$ km s$^{-1}$), is very old (ages$gtrsim 11$ Gyr), metal-rich ([Z/H]$sim0.2$ and [Na/Fe]$sim0.4$), and has a bottom-heavy IMF (with slope $Gamma_bsim2.4$). In the outer region, stars become increasingly hotter, younger, metal and sodium poorer, $alpha$-element richer, and the IMF slope becomes Chabrier-like. The multiple structural components in NGC 3311 confirm the predictions from the two-phase formation scenario for NGC 3311. Interestingly, the outer stellar population has an overabundant [$alpha$/Fe], most likely because NGC 3311, located at the center of the galaxy cluster, accreted stars from rapidly quenched satellites.[Abridged]
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