  We calculate the $\Lambda_b \to \Lambda_c \ell \nu$ form factors and decay
rates for all possible $b\to c \ell\bar\nu$ four-Fermi interactions beyond the
Standard Model, including nonzero charged lepton masses and terms up to order
$\alpha_s\, \Lambda_\text{QCD}/m_{c,b}$ and $\Lambda_\text{QCD}^2/m_c^2$ in the
heavy quark effective theory. At this order, we obtain model independent
predictions for semileptonic $\Lambda_b \to \Lambda_c$ decays in terms of only
two unknown sub-subleading Isgur-Wise functions, which can be determined from
fitting LHCb and lattice QCD data. We thus obtain model independent results for
$\Lambda_b\to \Lambda_c\ell\bar\nu$ decays, including predictions for the ratio
$R(\Lambda_c) = {\cal B}(\Lambda_b\to \Lambda_c \tau\bar\nu) / {\cal
B}(\Lambda_b\to \Lambda_c \mu\bar\nu)$ in the presence of new physics, that are
more precise than prior results in the literature, and systematically
improvable with better data on the decays with $\mu$ (or $e$) in the final
state. We also explore tests of factorization in $\Lambda_b \to \Lambda_c\pi$
decays, and emphasize the importance of measuring at LHCb the double
differential rate $d^2\Gamma(\Lambda_b\to\Lambda_c\ell\bar\nu) / (d q^2\,
d\cos\theta)$, in addition to the $q^2$ spectrum.
