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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?

My thoughts to ii) are: If a power series diverges for some element $a \in k$, it has to diverge for every element of the same norm. Still I found papers where people consider non-archimedean meromorphic functions with a discrete set of singularities. I am quite confused right now. Thank you for disconfusing me. Edit: The confusion was fixed (see comments).

Also, it's the kind of question that I would guess a whole industry of mathematicians are working on. If it's not already solved. If it is solved, my question is covered by i) and ii). If it is not already solved, consider iii) and iv):

iii) Who is known to work on this problem?

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

Edit: Jérôme answered ii) and probably iii) in the comments.

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?

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?

My thoughts to ii) are: If a power series diverges for some element $a \in k$, it has to diverge for every element of the same norm. Still I found papers where people consider non-archimedean meromorphic functions with a discrete set of singularities. I am quite confused right now. Thank you for disconfusing me.

Also, it's the kind of question that I would guess a whole industry of mathematicians are working on. If it's not already solved. If it is solved, my question is covered by i) and ii). If it is not already solved, consider iii) and iv):

iii) Who is known to work on this problem?

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

Edit: Jérôme answered ii) and probably iii) and iv) in the comments.

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?

My thoughts to ii) are: If a power series diverges for some element $a \in k$, it has to diverge for every element of the same norm. Still I found papers where people consider non-archimedean meromorphic functions with a discrete set of singularities. I am quite confused right now. Thank you for disconfusing me. Edit: The confusion was fixed (see comments).

Also, it's the kind of question that I would guess a whole industry of mathematicians are working on. If it's not already solved. If it is solved, my question is covered by i) and ii). If it is not already solved, consider iii) and iv):

iii) Who is known to work on this problem?

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

Edit: Jérôme answered ii) and probably iii) in the comments.