  In this work, the lepton-number-violating processes in $|\Delta L|=2$ decays
of $\Lambda_b^0$ baryon, $\Lambda_b^0 \to p \pi^+\mu^- \mu^-$ and $\Lambda_b^0
\to \Lambda_c^+ \pi^+\mu^- \mu^-$, are investigated for the first time, via an
intermediate on-shell Majorana neutrino $N$ with a mass in the GeV scale. We
explore the experimental sensitivity of these dimuon channels at the LHCb and
CMS experiments, in which heavy neutrino lifetimes in the accessible ranges of
$\tau_N = [1, 100, 1000]$ ps are considered. For a integrated luminosity
collected of 10 and 50 fb${}^{-1}$ at the LHCb and 30, 300 and 3000 fb${}^{-1}$
at the CMS, we found significant sensitivity on branching fractions of the
order ${\rm BR}(\Lambda_b^0 \to p \pi^+\mu^- \mu^-) \lesssim
\mathcal{O}(10^{-9} - 10^{-8})$ and ${\rm BR}(\Lambda_b^0 \to \Lambda_c^+
\pi^+\mu^- \mu^-) \lesssim \mathcal{O}(10^{-8} - 10^{-7})$. Exclusion regions
on the parameter space ($m_N,|V_{\mu N}|^{2}$) associated with the heavy
neutrino are presented and compared with those from $K^- \to \pi^+\mu^-\mu^-$
(NA48/2) and $B^- \to \pi^+\mu^-\mu^-$ (LHCb) as well as by different search
strategies such as NA3, CHARMII, NuTeV, Belle, and DELPHI.
