We prove the almost sure invariance principle for martingales with stationary ergodic differences taking values in a separable $2$-smooth Banach space (for instance a Hilbert space). A compact law of the iterated logarithm is established in the case of stationary differences of \emph{reverse} martingales. Then, we deduce the almost sure invariance principle for stationary processes under the Hannan condition; and a compact law of the iterated logarithm for stationary processes arising from non-invertible dynamical systems. Those results for stationary processes are new, even in the real valued case. We also obtain the Marcinkiewicz-Zygmund strong law of large numbers for stationary processes with values in some smooth Banach spaces.