  In this paper we develop a stochastic integration theory for processes with
values in a quasi-Banach space. The integrator is a cylindrical Brownian
motion. The main results give sufficient conditions for stochastic
integrability. They are natural extensions of known results in the Banach space
setting. We apply our main results to the stochastic heat equation where the
forcing terms are assumed to have Besov regularity in the space variable with
integrability exponent $p\in (0,1]$. The latter is natural to consider for its
potential application to adaptive wavelet methods for stochastic partial
differential equations.
