Is it possible to extend a diffeomorphism of $[0,1]^n$ to a diffeomorphism of a compact infinite-dimensional manifold?

For example, we can always extend a diffeomorphism $f$ of $[0,1]^n$ to a diffeomorphism $g$ of $[0,1]^m$ for $m >n$ by $g(x_1,...,x_n,...,x_m)=(f(x_1,...,x_n),x_{n+1},...,x_m)$.

Is there an analogous method from $[0,1]^n$ to a compact infinite-dimensional manifold?