  New physics close to the electroweak scale is well motivated by a number of
theoretical arguments. However, colliders, most notably the Large Hadron
Collider (LHC), have failed to deliver evidence for physics beyond the Standard
Model. One possibility for how new electroweak-scale particles could have
evaded detection so far is if they carry only electroweak charge, i.e. are
color neutral. Future $e^+e^-$ colliders are prime tools to study such new
physics. Here, we investigate the sensitivity of $e^+e^-$ colliders to scalar
partners of the charged leptons, known as sleptons in supersymmetric extensions
of the Standard Model. In order to allow such scalar lepton partners to decay,
we consider models with an additional neutral fermion, which in supersymmetric
models corresponds to a neutralino. We demonstrate that future $e^+e^-$
colliders would be able to probe most of the kinematically accessible parameter
space, i.e. where the mass of the scalar lepton partner is less than half of
the collider's center-of-mass energy, with only a few days of data. Besides
constraining more general models, this would allow to probe some well motivated
dark matter scenarios in the Minimal Supersymmetric Standard Model, in
particular the incredible bulk and stau co-annihilation scenarios.
