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In this work, we explore the signals of an $S_3$-symmetric two Higgs doublet model with two generations of vector-like leptons (VLLs) at the proposed International Linear Collider (ILC). The lightest neutral component of the VLL in this model provides a viable dark matter (DM) candidate satisfying the current relic density data as well as circumventing all direct and indirect DM search constraints. Some representative benchmark points have been selected with low, medium and high DM masses, satisfying all theoretical and experimental constraints of the model and constraints coming from the DM sector. The VLLs (both neutral and charged) will be produced in pair leading to multi-lepton and multi-jet final states. We show that the ILC will prove to be a much more efficient and useful machine to hunt for such signals compared to LHC. Using traditional cut-based analysis as well as sophisticated multivariate analysis, we perform a detailed analysis of some promising channels containing mono-lepton plus di-jet, di-lepton, four-lepton and four jets along with missing transverse energy in the final state at 1 TeV ILC.