  This document summarises proposed searches for new physics accessible in the
heavy-ion mode at the CERN Large Hadron Collider (LHC), both through hadronic
and ultraperipheral $\gamma\gamma$ interactions, and that have a competitive
or, even, unique discovery potential compared to standard proton-proton
collision studies. Illustrative examples include searches for new particles --
such as axion-like pseudoscalars, radions, magnetic monopoles, new long-lived
particles, dark photons, and sexaquarks as dark matter candidates -- as well as
new interactions, such as non-linear or non-commutative QED extensions. We
argue that such interesting possibilities constitute a well-justified
scientific motivation, complementing standard quark-gluon-plasma physics
studies, to continue running with ions at the LHC after the Run-4, i.e. beyond
2030, including light and intermediate-mass ion species, accumulating
nucleon-nucleon integrated luminosities in the accessible fb$^{-1}$ range per
month.
