An example of a free-by-cyclic group that is not CAT(0) was [given][1] by Gersten.  It is constructed from the automorphism of $F_3\cong\langle a,b,c\rangle$ that sends

$a\mapsto a,~b\mapsto ba,~ c\mapsto ca^2~.$

The idea of the proof is to think about translation lengths and flats in any CAT(0) space on which it acts.  As $\langle a,t\rangle\cong\mathbb{Z}^2$, it stabilises some flat.  But $t$, $at$ and $a^2t$ are all conjugate, so have the same translation lengths.  A little thought shows that this is impossible in a flat.

Note that, in many respects, (fg free)-by-cyclic groups are difficult to distinguish from CAT(0) groups.  For instance, they have [quadratic isoperimetric inequality][2].


  [1]: http://www.ams.org/mathscinet/search/publdoc.html?pg1=IID&s1=190275&vfpref=html&r=18&mx-pid=1195719
  [2]: http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=AUCN&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=bridson%252C%2520m*&s5=groves&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=2&mx-pid=2590896