  We construct the first example of a finitely generated group which has
Serre's property (FA) (i.e., whenever it acts on a simplicial tree it fixes a
vertex), but admits a fixed point-free action on an $\mathbb{R}$-tree with
finite arc stabilizers. We also give a short and elementary construction of
finitely generated groups that have property (FA) but do not have
(F$\mathbb{R}$).
