  The main goal of this article is to prove the existence of a random attractor
for a stochastic evolution equation driven by a fractional Brownian motion with
$H\in (1/2,1)$. We would like to emphasize that we do not use the usual
cohomology method, consisting of transforming the stochastic equation into a
random one, but we deal directly with the stochastic equation. In particular,
in order to get adequate a priori estimates of the solution needed for the
existence of an absorbing ball, we will introduce stopping times to control the
size of the noise. In a first part of this article we shall obtain the
existence of a pullback attractor for the non-autonomous dynamical system
generated by the pathwise mild solution of an nonlinear infinite-dimensional
evolution equation with non--trivial H\"older continuous driving function. In a
second part, we shall consider the random setup: stochastic equations having as
driving process a fractional Brownian motion with $H\in (1/2,1)$. Under a
smallness condition for that noise we will show the existence and uniqueness of
a random attractor for the stochastic evolution equation.
