A large variety of new physics models suggest that the rates for lepton flavour violating $b$-hadron decays may be much higher than predicted in the Standard Model, which leads to a high interest in the search for such decays. This thesis presents the search for the lepton-flavour violating decays $B^0\to e^{\pm}\mu^{\mp}$ and $B^0_s\to e^{\pm}\mu^{\mp}$ with the data sample collected by the LHCb experiment between 2016 and 2018, which corresponds to an integrated luminosity of 5.4 fb$^{-1}$ of proton-proton collisions. The analysis is performed while blinding the signal mass region, to avoid any bias introduced by the experimenter. The $B^0\to e^{\pm}\mu^{\mp}$ and $B^0_s\to e^{\pm}\mu^{\mp}$ branching fractions are measured with respect to the high statistics $B^+\to (J/\psi \to \mu^+ \mu^-)K^+$ decay, with a fit to the invariant $e^{\pm}\mu^{\mp}$ mass distribution in six independent data samples characterized by different background levels and signal mass resolutions due to electron bremsstrahlung. A dedicated selection is used to reduce background from random combinations and specific physics decays with a multivariate analysis and particle identification requirements, respectively. Efficiencies are determined from simulation, corrected for mismodeling effects with data-driven methods, and validated by measuring $r_{J/\psi} = \mathcal{B}(J/\psi \to \mu^+\mu^-)/\mathcal{B}(J/\psi \to e^+e^-)$. The main physics backgrounds are identified to arise from $B^0$ decays to the $K^+\pi^-$, $\pi^+\pi^-$, $\pi^+e^-\bar{\nu}_e$ and $\pi^+\mu^-\bar{\nu}_{\mu}$ final states, and taken into account in the mass fit; the yields of the two components peaking in the signal mass window, $B^0\to K^+\pi^-$ (12.2 events) and $B^0\to \pi^+\pi^-$ (7.7 events), are estimated from simulation and validated with two independent data-driven methods. Pseudo-experiments are performed to test the fit stability, study potential biases, and estimate the sensitivity of the analysis. In absence of signals and without any systematic uncertainty considered, the average expected upper limits at 95\%(90\%) confidence level are estimated to be $\mathcal{B}(B^0\to e^{\pm}\mu^{\mp})< 7.3 (5.9)\cdot 10^{-10}$, and $\mathcal{B}(B^0_{s}\to e^{\pm}\mu^{\mp})<2.3(1.8)\cdot 10^{-9}$ or $<1.8(1.5)\cdot 10^{-9}$ assuming the $B^0_s$ decay amplitude is completely dominated by the light (short-lived) or heavy (long-lived) mass eigenstate of the $B^0_s -\overline{B}^0_s$ system, respectively. This is an improvement in sensitivity by a factor 1.6 for $B^0\to e^{\pm}\mu^{\mp}$ and 2.8 for $B^0_{s}\to e^{\pm}\mu^{\mp}$ (when dominated by the heavy eigenstate), with respect to the previous best expected limits published by LHCb with the 2011 and 2012 data. In addition, this thesis presents the quality assurance process of silicon photomultiplier detectors used for the Scintillating Fibre (SciFi) tracker of the recent LHCb detector upgrade.