Let $n$ be a positive integer, and let $\mathbf{v}$ be a non-zero vector in $\mathbb{R}^n$.
Let $\; p : [0,1] \to \mathbb{R}^n \;$ be $C^1$ and such that $p'$ is nowhere zero.
Does there always exist $\; f : ([0,1] \times \mathbb{R}^n) \to \mathbb{R}^n \;$ such that $f\hspace{0.01 in}$ is $C^1$ and for all members $t$ of $[0,1]$, $\; (\mathbf{x} \in \mathbb{R}^n) \mapsto f\hspace{0.01 in}(\langle t,\mathbf{x} \rangle) \;$ is a $C^1$ diffeomorphism of $\mathbb{R}^n$ and $\; f\hspace{0.01 in}(\langle 1,p(t)\rangle) = t\cdot \mathbf{v} \;$ ?

(That would be the $C^1$ version of an ambient isotopy.
I already know about the Fox-Artin arc, and I can see that it can 'easily' be
made to be discontinuously differentiable with a nowhere zero derivative.)