  In this article we introduce cylindrical fractional Brownian motions in
Banach spaces and develop the related stochastic integration theory. Here a
cylindrical fractional Brownian motion is understood in the classical framework
of cylindrical random variables and cylindrical measures. The developed
stochastic integral for deterministic operator valued integrands is based on a
series representation of the cylindrical fractional Brownian motion, which is
analogous to the Karhunen-Lo\`eve expansion for genuine stochastic processes.
In the last part we apply our results to study the abstract stochastic Cauchy
problem in a Banach space driven by cylindrical fractional Brownian motion.
