Let $B$ be a Banach space. Let $\{c_{n}\}_{n\in\mathbb{N}}$ be a sequence of smooth curves $c_{n}\colon \mathbb{R}\rightarrow B$ such that $c_{n}(0)=x$, and such that the sequence $\{v_{n}\}_{n\in\mathbb{N}}$ of tangent vectors $v_{n}=\left.\frac{\mathrm{d} c_{n}}{\mathrm{d}t}\right|_{t=0}$ convergences to $v\in B$ uniformly.
What can we say about the convergence of $\{c_{n}\}_{n\in\mathbb{N}}$? Are there reasonable assumptions to ensure it converges?
Yo can’t expect a lot...
Take the example of a family of circles in the real plane all passing through the origin and tangent to the $x$-axis, with $v_n=v \neq 0$ for all $n \in \mathbb N$. Also suppose that the radius of $c_n$ is equal to $n$.
You don’t have uniform convergence (neither pointwise by the way).