Small-animal PET systems are now striving for sub-millimetre resolution. Current systems based upon PSPMTs and finely pixellated scintillators can be pushed to higher resolution, but at the expense of other performance parameters and a rapidly escalating cost. Moreover, depth of interaction (DOI) information is usually difficult to assess in such systems, even though this information is highly desirable to reduce the parallax error, which is often the dominant error for such high-resolution systems. In this study we propose a high-resolution detector head for a small-animal PET imaging system with intrinsic DOI information. Instead of a pixellated scintillator, our design is based upon the classic Anger camera principle, i.e. the head is constructed of modular layers each consisting of a continuous slab of scintillator, viewed by a new type of compact silicon photodetector. The photodetector is the recently developed silicon photomultiplier (SiPM) that as well as being very compact has many other attractive properties: high gain at low bias voltage, excellent single-photoelectron resolution and fast timing. A detector head of about 4 x 4 cm2 in area is proposed, constructed from three modular layers of the type described above. We perform a simulation study, using the Monte Carlo simulation package Geant4. The simulation results are used to optimize the geometry of the detector head and characterize its performance. Additionally, hit estimation algorithms are studied to determine the interaction position of annihilation photons correctly over the whole detector surface. The resulting detector has a nearly uniform efficiency for 511 keV photons of approximately 70% and an intrinsic spatial resolution of less than approximately 0.4 mm full width at half maximum (fwhm).