We present a detailed analysis of Swift multiwavelength observations of GRB 070110 and its remarkable afterglow. The early X-ray light curve, interpreted as the tail of the prompt emission, displays a spectral evolution already seen in other gamma-ray bursts. The optical afterglow shows a shallow decay up to similar to 2 days after the burst, which is not consistent with standard afterglow models. The most intriguing feature is a very steep decay in the X-ray flux at similar to 2 x 10 super(-1) s after the burst, ending an apparent plateau. The abrupt drop of the X-ray light curve rules out an external shock as the origin of the plateau in this burst and implies long-lasting activity of the central engine. The temporal and spectral properties of the plateau phase point toward a continuous central engine emission rather than the episodic emission of X-ray flares. We suggest that the observed X-ray plateau is powered by a spinning-down central engine, possibly a millisecond pulsar, which dissipates energy at an internal radius before depositing energy into the external shock.