Some conic bundles that are not birationally trivial do the job. For explicit examples, see 

<cite authors="Matsuki, Kenji">_Matsuki, Kenji_, Introduction to the Mori program, Universitext. New York, NY: Springer (ISBN 0-387-98465-8/hbk). xxiii, 478&nbsp;p. (2002). [ZBL0988.14007](https://zbmath.org/?q=an:0988.14007),</cite> 

pages 143-148.