The future evolution of the Greenland ice sheet (GrIS) depends on the intensity and the speed of climate change. By applying different rates of temperature change in a state-of-the-art comprehensive ice-sheet model coupled to a regional energy-moisture balance atmospheric model, oscillations in the total ice-sheet volume are found under warming magnitudes between 1.0 and 1.3 K above present-day temperatures. These are located in the northwestern drainage basin of the GrIS and are due to two ice streams which alternate between fast and slow basal velocities, manifesting in a build-up/surge variability. These ice streams interact due to their spatial proximity, resulting in irregular periodicity. The ice streams appear in a region where tipping of the entire GrIS begins, leading the oscillations to affect the tipping behaviour. These oscillations directly impact the time it takes before the ice sheet collapses at a given external forcing magnitude by hundreds of thousands of years for an ensemble of rates of forcing and initial conditions. These long tipping times are proposed to be due to chaotic transients. Our results suggest that ice-stream oscillations are a potential source of internal chaotic variability in ice sheets that affect tipping behaviour, thereby complicating prospects of anticipating such a tipping.