  The exclusive weak decay of $\Lambda_b \to \Lambda \gamma$ and $\Lambda_b \to
\Lambda l^{+} l^{-}$ are investigated in the Applequist-Cheng-Dobrescu model,
which is an extension of the standard model in presence of universal extra
dimensions. Employing the transition form factors obtained in the light-cone
sum rues, we analyze how the invariant mass distribution, forward-backward
asymmetry and polarization asymmetry of $\Lambda$ baryon of these decay modes
can be used to constrain the only one additional free parameter with respect to
the standard model, namely, the radius $R$ of the extra dimension. Our results
indicate that the Kaluza-Klein modes can lead to approximately 25% suppression
of the branching ratio of $\Lambda_b \to \Lambda \gamma$, however, their
contributions can bring about 10% enhancement to the decay rate of $\Lambda_b
\to \Lambda l^{+} l^{-}$. It is shown that the zero-position of
forward-backward asymmetry of $\Lambda_b \to \Lambda \mu^{+} \mu^{-}$ is
sensitive to the compactification parameter $R$ in this scenario, while the
measurement of polarizations of $\Lambda$ baryon in the $\Lambda_b$ decays is
not suitable to provide some valuable information for the universal extra
dimension model.
