  A theory for laser-driven electron dynamics and high-harmonic generation in
bulk solids with two lattice sites per unit cell of arbitrary dimension is
formulated. In tight-binding approximation, such solids can be described by
$2\times 2$ Bloch-Hamiltonians. Our theory is able to fully capture topological
effects in high-harmonic generation by such systems because no simplifications
beyond tight-binding, dipole approximation, and negligible depletion of the
valence band are made. An explicit, analytical expression for the electron
velocity is given. Exemplarily, the theory is applied to the
Su-Schrieffer-Heeger chain and the Haldane model in strong laser fields.
