  This work is the continuation of the recent paper \cite{D2} devoted to the
density-dependent incompressible Euler equations. Here we concentrate on the
well-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in the
set of Lipschitz continuous functions, a functional framework which contains
the particular case of H\"older spaces and of the endpoint Besov space
$B^1_{\infty,1}.$ For such data and under the nonvacuum assumption, we
establish the local well-posedness and a continuation criterion in the spirit
of that of Beale, Kato and Majda in \cite{BKM}.
  In the last part of the paper, we give lower bounds for the lifespan of a
solution. In dimension two, we point out that the lifespan tends to infinity
when the initial density tends to be a constant. This is, to our knowledge, the
first result of this kind for the density-dependent incompressible Euler
equations.
