Is it true that the space $C_c^{\infty}([0,T];V)$ is dense in $C_c^{1}([0,T];V)$? These are compactly supported functions that are $V$ valued, where $V$ is a Banach or Hilbert space.

As Andre suggests: Convolution with smooth bump function with very small support will give you an approximation by a smooth function which however need not have support in $[0,T]$. However, you may first squeeze the support of the given function you want to approximate in order to make its support a compact subset of $(0,T)$. Then the support of the convolution stays in $[0,T]$ if the support of the bump function is small enough. 

