Can a tangent vector extend to a vector field?

Question

Let $M$ be a smooth manifold and $p\in M$. I would like to know whether any tangent vector $X_p \in T_pM$ extends to a vector field over $M$.

If so is it unique? How can I construct it?

Answer 1

First notice that this problem is easy when you're in $\mathbb{R}^n$. Then use a small coordinate patch and a bump function to extend the vector field as you wish.