Weierstrass's function is the real part of
$$\sum_{n=0}^\infty a^nz^{b^n},\quad |z|\leq 1,$$
where $b\geq 2$ is an integer, and $a<1$. It was studied by complex analysis
in G. H. Hardy, in a series of papers, for example in TAMS, 17, 301-325 (1916).
Interesting examples can be of two kinds: a) useful examples, I mean useful in other areas of science and mathematics, and b) counterexamples, specially constructed to disprove some conjectures, or to show that certain assumptions
cannot be removed from some theorems, like the Weierstrass's function.
I will address only useful functions below.
Useful examples of analytic functions are solutions of various functional equations, first of all those related to differential equations.
Whittaker-Watson, vol. II is a good source for special functions which were
studied by the end of 19th century (Gamma, hypergeometric family (incl. Bessel, Airy, Weber, classical orthogonal polynomials etc.), Matieu and Lame functions, elliptic and theta
and Riemann zeta.
To this collection solutions of Painleve and Heun equations were added in 20th century, but this material is by far too advanced and cannot be addressed in a general complex variables course.
There are interesting functions with non-trivial singularities which arise in
holomorphic dynamics as solutions of functional equations of Schroeder, Abel and Poincare.
These are more easily accessible, and some of them can be included in a general course. (This is to address "interesting natural domains" in your wish list. They were actually studied for the first time by Fatou because of their funny
natural domains). Poincare functions give the famous Fatou-Bieberbach domains.
Other class of examples with interesting natural domains are automorphic functions, it is not difficult to give some simple examples, related to Fuchsian ot Schottky groups.
There are some other interesting and useful solutions of functional equations, my favorite one is the deformed exponential which solves
$$f'(z)=f(az),\quad f(0)=1,$$
where $a$ is a complex parameter, $|a|\leq 1$, but properties of this function,
for non-real $a$
except the simplest ones are still a complete mystery. For $a\in(-1,1)$ it
is well understood, and can serve as an example in a course of complex variables. It is related to certain generating functions in problems of physics and graph theory.
Of multi-valued functions, of course algebraic ones and Abelian integrals are the most important examples.
Ahlfors's textbook on complex variables covers the "minimal set" of most important functions listed above (Gamma, hypergeometric, elliptic, modular function and zeta).
Best Answer
Benedetto has a textbook that discusses basic $p$-adic analysis, although his aim is to study $p$-adic dynamics. And it's for a single variable. But might be a good place to get some information.
Dynamics in One Non-Archimedean Variable, Robert L. Benedetto, Graduate Studies in Mathematics, Volume 198, 2019, American Mathematical Society
I'll also mention that these days a lot of $p$-adic analysis is done on Berkovich space, rather than on $\mathbb Q_p$ or $\mathbb C_p$. An introduction to analysis on the Berkovich line can be found in the book of Baker and Rumely.
Potential Theory and Dynamics on the Berkovich Projective Line, Matthew Baker, Robert Rumely, Mathematical Surveys and Monographs Volume 159, 2010, American Mathematical Society.