  Driven by the recent experimental hints of lepton-flavor-universality
violation in the bottom-quark sector, we consider a simple extension of the
Standard Model (SM) with an additional vector leptoquark $V_{\rm LQ}({\bf
3},{\bf 1},2/3)$ and a scalar diquark $S_{\rm DQ}({\bf 6},{\bf 1},4/3)$ under
the SM gauge group $SU(3)_c\times SU(2)_L\times U(1)_Y$, in order to
simultaneously explain the $b \to s \ell^+ \ell^-$ (with $\ell=e,\mu$) and $b
\to c l^- \bar \nu_l$ (with $l=e,\mu,\tau$) flavor anomalies, as well as to
generate small neutrino masses through a two-loop radiative mechanism. We
perform a global fit to all the relevant and up-to-date $b \to s \ell^+ \ell^-$
and $b \to c l^- \bar \nu_l$ data under the assumption that the leptoquark
couples predominantly to second and third-generation SM fermions. We then look
over the implications of the allowed parameter space on lepton-flavor-violating
$B$ and $\tau$ decay modes, such as $B_s \to l^+_i l^-_j, \ B \to K^{(*)} l^+_i
l^-_j, \ B_s \to \phi l^+_i l^-_j$, $\Upsilon(nS) \to \mu \tau$ and $\tau \to
\mu \gamma$, $\tau \to \mu \phi (\eta^{(\prime)})$, respectively. Minimally
extending this model by adding a fermion singlet $\chi({\bf 1},{\bf 1},0)$ also
explains the ANITA anomalous upgoing events. Furthermore, we provide
complementary constraints on leptoquark and diquark couplings from high-energy
collider and other low-energy experiments to test this model.
