  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 $R\lesssim 2.0$ 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 $\Gamma_b\sim2.4$). 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]
