  We compute the momentum diffusion coefficients of heavy quarks,
$\kappa_\parallel$ and $\kappa_\perp$, in a strong magnetic field $B$ along the
directions parallel and perpendicular to $B$, respectively, at the leading
order in QCD coupling constant $\alpha_s$. We consider a regime relevant for
the relativistic heavy ion collisions, $\alpha_s eB\ll T^2\ll eB$, so that
thermal excitations of light quarks are restricted to the lowest Landau level
(LLL) states. In the vanishing light-quark mass limit, we find
$\kappa_\perp^{\rm LO}\propto \alpha_s^2 T eB$ in the leading order that arises
from screened Coulomb scatterings with (1+1)-dimensional LLL quarks, while
$\kappa_\parallel$ gets no contribution from the scatterings with LLL quarks
due to kinematic restrictions. We show that the first non-zero leading order
contributions to $\kappa_\parallel^{\rm LO}$ come from the two separate
effects: 1) the screened Coulomb scatterings with thermal gluons, and 2) a
finite light-quark mass $m_q$. The former leads to $\kappa_\parallel^{\rm
LO,\,gluon} \propto \alpha_s^2 T^3$ and the latter to $\kappa_\parallel^{\rm
LO,\,massive}\propto \alpha_s (\alpha_s eB)^{1/2} m_q^2$. Based on our results,
we propose a new scenario for the large value of heavy-quark elliptic flow
observed in RHIC and LHC. Namely, when $\kappa_\perp\gg\kappa_\parallel$, an
anisotropy in drag forces gives rise to a sizable amount of the heavy-quark
elliptic flow even if heavy quarks do not fully belong to an ellipsoidally
expanding background fluid.
