We can represent subclass of linear time invariant (LTI) systems with State Space Representation:
$$\dot X = AX + BU,$$ $$Y = CX + DU.$$
Also, nonlinear systems are formulated with generalized State Space Representation where the functions could be nonlinear:
$$\dot X=f(X,U,t),$$ $$Y = g(X,U,t).$$
The question is, what is their representation power? Can we formulate any system in State Space Model?
It depends on how much you "generalize" the state space representation when dealing with non-linear dynamical systems.
$$ \dot X = f\left( {X,U,t} \right)\quad \quad Y = g\left( {X,U,t} \right)$$
The general consensus is that in the above generalizations derivatives of the input $U$ are absent. However, one may indeed have systems of the form $ \dot x = f\left( {x,\dot u} \right) $ and some work has been done by Freedman and Willems on conditions under which they can be cast into the form $ \dot z = g\left( {z,u} \right);x = h\left( {z,u} \right)$.
It is known that given an arbitrary system satisfying the form of generalized controller canonical representation obtaining a state-space representation from it is a hard problem. More so if the representation desired is affine in $U$.