We present observations of the X-ray afterglow of GRB 050408, a gamma-ray burst discovered by HETE-II. Swift began observing the field 42 min after the burst, performing follow-up over a period of 38 d (thus spanning three decades in time). The X-ray light curve showed a steepening with time, similar to many other afterglows. However, the steepening was unusually smooth, over the duration of the XRT observation, with no clear break time. The early decay was too flat to be described in terms of standard models. We therefore explore alternative explanations, such as the presence of a structured afterglow or of long-lasting energy injection into the fireball from the central GRB engine. The lack of a sharp break puts constraints on these two models. In the former case, it may indicate that the angular energy profile of the jet was not a simple power law, while in the second model it implies that injection did not stop abruptly. The late decay may be due either to a standard afterglow (that is with no energy injection), or to a jetted outflow still being refreshed. A significant amount of absorption was present in the X-ray spectrum, corresponding to a rest-frame hydrogen column density NH = 1.2-0.3+0.4 × 1022 cm-2, indicative of a dense environment.