  We show that a minor refinement of the Bourgain-Rosenthal construction of a
Banach space without the Radon-Nikodym property which contains no bounded
$\delta$-trees yields a space with the Daugavet property and the Schur
property. Using this example we answer some open questions on the structure of
such spaces; in particular we show that the Daugavet property is not inherited
by ultraproducts.
