  The measurement of a large like-sign dimuon asymmetry $A^b_{SL}$ by the D0
experiment at the Tevatron departs noticeably from Standard Model expectations
and it may be interpreted as a hint of physics beyond the Standard Model
contributing to $\Delta B\neq 0$ transitions. In this work we analyse how the
natural suppression of $A^b_{SL}$ in the SM can be circumvented by New Physics.
We consider generic Standard Model extensions where the charged current mixing
matrix is enlarged with respect to the usual $3\times 3$ unitary
Cabibbo-Kobayashi-Maskawa matrix, and show how, within this framework, a
significant enhancement over Standard Model expectations for $A^b_{SL}$ is
easily reachable through enhancements of the semileptonic asymmetries
$A^d_{SL}$ and $A^s_{SL}$ of both $B^0_d$ - $\bar B^0_d$ and $B^0_s$ - $\bar
B^0_s$ systems. Despite being insufficient to reproduce the D0 measurement,
such deviations from SM expectations may be probed by the LHCb experiment.
