Let's take a disk $D^2 \subset \mathbb{R}^2$ with $n$ holes ($n = 0, 1, ...$).
In case $n = 0, 1$ it's clear how to extend any non-zero (i.e. with no singular points) vector field from $S^1$ to disk in first case and a ring in the second.
I expect that in case $n > 1$ it's also possible to extend vector field initially set on each of $S^1$ of inner holes to the whole $D^2$.
More formally, let $\{S^1\}_{i=1..n}$ - borders of holes in $D^2$ and $v|_{S^1 _{i}}$ are provided. How to extend $v$ to the whole $D^2$?