We present the first fully non-linear evolutions of binary neutron star mergers in a moving-punctures approach in Einstein–scalar–Gauss–Bonnet gravity. We study both linear and quadratic-type couplings between the scalar and the Gauss-Bonnet invariant, and uncover new post-merger phenomena. These include an enhancement of the prompt collapse of a long-lived hyper-massive neutron star remnant and cases where the remnant develops a scalar configuration due to different scalarisation instabilities. This study initiates the exploration of beyond-General-Relativistic effects enhanced by the non-linear dynamics of the neutron star’s fluid.