  We study a scalar field theory with a non-standard kinetic term minimally
coupled to gravity. We establish the existence of compact boson stars, that is,
static solutions with compact support of the full system with self-gravitation
taken into account. Concretely, there exist two types of solutions, namely
compact balls on the one hand, and compact shells on the other hand. The
compact balls have a naked singularity at the center. The inner boundary of the
compact shells is singular, as well, but it is, at the same time, a Killing
horizon. These singular, compact shells therefore resemble black holes.
