This thesis presents a search for a beyond the Standard Model (SM) heavy neutral lepton (HNL), which does not interact via any of the SM interactions. This hypothetical particle has been proposed by multiple theories as an extension to the SM, and it could solve one or several experimental puzzles of the modern high energy physics. The search is done using the dataset collected by the LHCb experiment during the years 2016-2018, corresponding to an integrated luminosity of $5.0~{\rm fb}^{-1}$. Its signature consists in a heavy neutrino decaying to $\mu^\pm\pi^\mp$ final state, while the heavy neutrino is produced in $B$ meson decays to $\mu^+N$ or $\mu^+NX$, where $X$ is not reconstructed. Results of this research work are presented as the limit on the mixing parameter $|U_{\mu}|^2$ between the HNL and the SM muon neutrino as a function of the HNL mass and lifetime that were considered. It improves the previous LHCb result by more than one order of magnitude, and is being competitive with the world-best limit.
Additionally, this manuscript describes a contribution to the development, commissioning and operation of the PLUME luminometer. This is a new subdetector of the LHCb experiment, which is dedicated to the luminosity measurement and is crucial for the operation of the detector.