Let $E$ be a Tate curve over a $p$-adic field $K$. Then there exists $q \in K^*$ with the valuation $v(q)>0$ such that $E(\overline{K})= \overline{K}^*/\left< q \right>$. So it is easy to see that the $n$-torsion points of $E$ have the form $$ E[n](\overline{K})=\dfrac{\left< q^{1/n}, \zeta_n \right>}{\left< q\right>}$$ where $\zeta_n$ a primitive nth root of unit and $q^{1/n}$ is a fixed nth root of $q$ in $\overline{K}$.
Do we have a description of the set of non-torsion points of $E$?
What you are asking for does not exist, and there is no reason to expect it should.
The set of all points of the Tate curve, or another elliptic curve, over a local field, or its algebraic closure, is large (it is an uncountable set) and smooth/homogeneous (it is the set of points of a smooth variety that is also a homogeneous space).
When you have a small (in this case, countable) subset in a large, homogeneous set, the complement rarely has a description simpler than the complement of some set inside some other set. For example, think of removing a bunch of points in the plane - there's not really a way to describe it other than taking the plane and removing these points, at least if you want to describe it with sufficient structure (e.g. as a Riemannian manifold) - as a topological space there can be alternative descriptions but they're not really any simpler.
The group of all points does have a description in terms of the formal group of the Tate curve (which is a generalization of the notion of Lie algebra), as the product of the integral points of the formal group with the $\mathbb Q/\mathbb Z$ and the roots of unity of the residue field, but considering the nontorsion points in this setup is again just taking the complement of a specified subset.
