  We show that many-body localization, which exists in tight-binding models, is
unstable in a continuum. Irrespective of the dimensionality of the system,
many-body localization does not survive the unbounded growth of the
single-particle localization length with increasing energy that is
characteristic of the continuum limit. The system remains delocalized down to
arbitrarily small temperature $T$, although its dynamics slows down as $T$
decreases. Remarkably, the conductivity vanishes with decreasing $T$ faster
than in the Arrhenius law. The system can be characterized by an effective
$T$-dependent single-particle mobility edge which diverges in the limit of
$T\to 0$. Delocalization is driven by interactions between hot electrons above
the mobility edge and the "bath" of thermal electrons in the vicinity of the
Fermi level.
