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.