GRB 050911, discovered by the Swift Burst Alert Telescope, was not seen 4.6 hr later by the Swift X-ray Telescope, making it one of the very few X-ray non-detections of a Gamma-Ray Burst (GRB) afterglow at early times. The gamma-ray light-curve shows at least three peaks, the first two of which (~T_0 - 0.8 and T_0 + 0.2 s, where T_0 is the trigger time) were short, each lasting 0.5 s. This was followed by later emission 10-20 s post-burst. The upper limit on the unabsorbed X-ray flux was 1.7 x 10^-14 erg cm^-2 s^-1 (integrating 46 ks of data taken between 11 and 18 September), indicating that the decay must have been rapid. All but one of the long bursts detected by Swift were above this limit at ~4.6 hr, whereas the afterglows of short bursts became undetectable more rapidly. Deep observations with Gemini also revealed no optical afterglow 12 hr after the burst, down to r=24.0 (5-sigma limit). We speculate that GRB 050911 may have been formed through a compact object (black hole-neutron star) merger, with the later outbursts due to a longer disc lifetime linked to a large mass ratio between the merging objects. Alternatively, the burst may have occured in a low density environment, leading to a weak, or non-existent, forward shock - the so-called naked GRB model.