Can Navier-Stokes equations be represented in the form
$\mathbf{x}'(t) = \mathbf{F}(\mathbf{x}(t))$ where values of $\mathbf{x}$ and $\mathbf{F}$ lie in some normed vector spaces over field $\mathbb{R}$?
Informally: Can Navier-Stokes equations be considered as equations on normed spaces over field $\mathbb{R}$?
The Navier Stokes equation is a partial differential equation which involves unknowns that depends on time $t$ and 3D space vector $x$, so if you want to write Navier-Stokes equation explicitly you can write it as a system of 3 PDEs supplemented with another equation which is the conservation of mass equation. But, it still a Partial differential equation so if you want to get the form: $$x'(t)=F(t,x(t)),$$ which is a differential equation you have first to use an approximation method. Methods like these are many, they aim to approximate a Partial differential equation by a differential equation in order to use the classic theory of differential equations, precisely the Cauchy-Lipschitz theorem to establish a result of existence and uniqueness. So the answer of your question is yes. You can write the Navier-Stokes equation in the form $$u'(t)=F(t,u(t)).$$