I would like as many examples as possible (with explanation) of the object mentioned in the title (for which $p$ does it exists ? How do you construct those generically ? Is there a 'classification' ?)
In the case $p=3$, $ \mathbb{Q}_3[X]/(X^3-3)$ is an example. But I have to say I'm not particularly interested by the peculiarities of the $p=3$ case.
I guess lots of people will say 'class field theory'. I have to admit I know nothing about that, but if you can mention which combinations of theorem I have to use to understand the example(s), I'm of course happy !
Actually, class field theory won't help you (at least not directly), since you're looking for non-abelian extensions!
Since you want examples, though, you should play with this wonderful database: https://math.la.asu.edu/~jj/localfields/
Away from $3$, your extension will be totally tamely ramified, and then it can always be written by taking the root of a uniformizer. At $3$, it's hard -- consult the tables!