What do we know about the types of singularities that a convergent power series over a non-archimedean field can have?

More specifically:

i) What types of essential singularities can occur?

ii) Are singularities isolated?

iii) Who is known to work on this problem? Edit: The answer is probably "nobody", see Jérôme's comments.

iv) In case the answer to iii) is "nobody": Why not?

Edit: Since I asked the question, I learned the following: Removable singularities are in fact removable (Bartenwerfer 1976).

I also have a new subquestion:

v) Develop an analytic function $f$ into a power series on some polydisc $D$. The function $f$ has a singularity on the boundary $\partial D$ of the disc. Assume the power series we got is an algebraic power series, i.e. an algebraic element over the ring of polynomials $k[T] \subset k[[T]]$. Is it true that $f$ has only finitely many poles and no essential singularities on $\partial D$?

vi) Does v) still hold if we work over a smooth affinoid algebra $A$ whose reduction $\tilde{A}$ is also smooth?

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