  The entanglement between spatial regions in an interacting Bose-Einstein
condensate is investigated using a quantum field theoretic formalism. Regions
that are small compared to the healing length are governed by a
non-relativistic quantum field theory in the vacuum limit, and we show that the
latter has vanishing entanglement. In the opposite limit of a region that is
large compared to the healing length, the entanglement entropy is like in the
vacuum of a relativistic theory where the velocity of light is replaced with
the velocity of sound and where the inverse healing length provides a natural
ultraviolet regularization scale. Besides the von Neumann entanglement entropy,
we also calculate R\'enyi entanglement entropies for a one-dimensional
quasi-condensate.
