  We study the capabilities of a muon collider experiment to detect
disappearing tracks originating when a heavy and electrically charged
long-lived particle decays via $X^+ \to Y^+ Z^0$, where $X^+$ and $Z^0$ are two
almost mass degenerate new states and $Y^+$ is a charged Standard Model
particle. The backgrounds induced by the in-flight decays of the muon beams
(BIB) can create detector hit combinations that mimic long-lived particle
signatures, making the search a daunting task. We design a simple strategy to
tame the BIB, based on a detector-hit-level selection exploiting timing
information and hit-to-hit correlations, followed by simple requirements on the
quality of reconstructed tracks. Our strategy allows us to reduce the number of
tracks from BIB to an average of 0.08 per event, hence being able to design a
cut-and-count analysis that shows that it is possible to cover weak doublets
and triplets with masses close to $\sqrt{s}/2$ in the 0.1-10 ns range. In
particular, this implies that a 10 TeV muon collider is able to probe thermal
MSSM higgsinos and thermal MSSM winos, thus rivaling the FCC-hh in that
respect, and further enlarging the physics program of the muon collider into
the territory of WIMP dark matter and long-lived signatures. We also provide
parton-to-reconstructed level efficiency maps, allowing an estimation of the
coverage of disappearing tracks at muon colliders for arbitrary models.
