We demonstrate that, for the baseline design of the CORE satellite mission, the polarized foregrounds can be controlled at the level required to allow the detection of the primordial cosmic microwave background (CMB) $B$-mode polarization with the desired accuracy at both reionization and recombination scales, for tensor-to-scalar ratio values of ${rgtrsim 5times 10^{-3}}$. We consider detailed sky simulations based on state-of-the-art CMB observations that consist of CMB polarization with $tau=0.055$ and tensor-to-scalar values ranging from $r=10^{-2}$ to $10^{-3}$, Galactic synchrotron, and thermal dust polarization with variable spectral indices over the sky, polarized anomalous microwave emission, polarized infrared and radio sources, and gravitational lensing effects. Using both parametric and blind approaches, we perform full component separation and likelihood analysis of the simulations, allowing us to quantify both uncertainties and biases on the reconstructed primordial $B$-modes. Under the assumption of perfect control of lensing effects, CORE would measure an unbiased estimate of $r=left(5 pm 0.4right)times 10^{-3}$ after foreground cleaning. In the presence of both gravitational lensing effects and astrophysical foregrounds, the significance of the detection is lowered, with CORE achieving a $4sigma$-measurement of $r=5times 10^{-3}$ after foreground cleaning and $60$% delensing. For lower tensor-to-scalar ratios ($r=10^{-3}$) the overall uncertainty on $r$ is dominated by foreground residuals, not by the 40% residual of lensing cosmic variance. Moreover, the residual contribution of unprocessed polarized point-sources can be the dominant foreground contamination to primordial B-modes at this $r$ level, even on relatively large angular scales, $ell sim 50$. Finally, we report two sources of potential bias for the detection of the primordial $B$-modes.[abridged]
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