We prove a sharp Hardy-type inequality for the Dirac operator. We exploit this inequality to obtain spectral properties of the Dirac operator perturbed with Hermitian matrix-valued potentials \(\mathbf V\) of Coulomb type: we characterise its eigenvalues in terms of the Birman-Schwinger principle and we bound its discrete spectrum from below, showing that the \emph{ground-state energy} is reached if and only if \(\mathbf V\) verifies some {rigidity} conditions. In the particular case of an electrostatic potential, these imply that \(\mathbf V\) is the Coulomb potential.