As Brian pointed out, the principal parts problem always has a solution on the open unit disk, and Lazard's 1962 article "Les zéros des fonctions analytiques d’une variable sur un corps valué complet" gives a nice proof which is also rather explicit.
Your problem has additional symmetries so one can be a bit more explicit, as follows.
Let H be the set of power series holomorphic on the open unit disk, let varphi:HtoH be the map defined by varphi(f)(X)=f((1+X)p−1) and let d be defined by d(f)(X)=(1+X)df/dX. Note that dvarphi=pvarphid. You can check that if you have a function f such that varphi(f)−pf=(1+X/2)log(1+X)/X, then f(0)neq0 and f(zetapn−1)=p−nf(0) so this answers your problem.
Therefore you need to be able to solve an equation of the form varphi(f)−pf=g. By taking d of both sides this gives varphi(df)−df=dg/p. Now you can solve an equation of the form varphi(a)−a=b if binXcdotH by writing a=sumngeq0varphin(b), which should converge in H for its Fréchet topology. We have d((1+X/2)log(1+X)/X)inXcdotH (this is what the (1+X/2) was put in for), and once you know df, you get f by integrating and adjusting the constant.
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