The zeroth law of turbulence states that, for fixed energy input into large-scale motions, the statistical steady state of a turbulent system is independent of microphysical dissipation properties. This behaviour, which is fundamental to nearly all fluid-like systems from industrial processes to galaxies, occurs because nonlinear processes generate smaller and smaller scales in the flow, until the dissipation – no matter how small – can thermalise the energy input. Using direct numerical simulations and theoretical arguments, we show that in strongly magnetised plasma turbulence such as that recently observed by the Parker Solar Probe spacecraft, the zeroth law is routinely violated. Namely, when such turbulence is ‘imbalanced’ – when the large-scale energy input is dominated by Alfvénic perturbations propagating in one direction (the most common situation in space plasmas) – nonlinear conservation laws imply the existence of a ‘barrier’ at scales near the ion gyroradius. This causes energy to build up over time at large scales. The resulting magnetic-energy spectra bear a strong resemblance to those observed in situ, exhibiting a sharp, steep kinetic transition range above and around the ion-Larmor scale, with flattening at yet smaller scales. The effect thus offers a possible solution to the decade-long puzzle of the position and variability of ion-kinetic spectral breaks in plasma turbulence. The existence of the ‘barrier’ also suggests that, how a plasma is forced at large scales (the imbalance) may have a crucial influence on thermodynamic properties such as the ion-to-electron heating ratio.