Future $c$-$\tau$ factories are natural places to study extensions of the Standard Model of particle physics (SM) with new long-lived feebly interacting particles light enough to be produced in electron-positron collisions. We investigate prospects of these machines in exploring such extensions emphasizing the role of polarized beams in getting rid of the SM irreducible background for the missing energy signature. We illustrate this on example of $c$-$\tau$ project in Novosibirsk, where the electron beam is designed to be polarized to achieve much higher sensitivity to hadronic resonances and $\tau$-leptons. We investigate models with hidden photons, with millicharged particles (fermions and scalars), with $Z'$ bosons and with axion-like particles. We find that the electron beam polarization of 80\% significantly improves the chances to observe the signal, especially with large statistics. We outline the regions of the model parameter space which can be reached at this factory in one year and in ten years of operation according with the scientific schedule of tuning the energy of colliding beams.
Comment: 12 pages, 21 figures; v2: including plots with limits from searches for visible modes if comparable to invisible, v3: replaced with final version