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We prove that every infinite-dimensional Banach space 
$X$ having a (not necessarily equivalent) real-analytic norm is real-analytic diffeomorphic to 
$X\,\backslash \,\left\{ 0 \right\}$. More generally, if $X$ is an infinite dimensional Banach space and 
$F$ is a closed subspace of $X$ such that there is a real-analytic seminorm on $X$ whose set of zeros is $F$, and 
$X/F$ is infinite-dimensional, then $X$ and 
$X\backslash F$ are real-analytic diffeomorphic. As an application we show the existence of real-analytic free actions of the circle and the 
$n$-torus on certain Banach spaces.
