Tetration and p-adic

[Ember Edison]

I would also love to see this in a p-adic sense! From my perspective it just smells like p-adic analysis so much. But I don't know enough Perhaps something on the lines of \(^\infty 2\) is a repeating fraction in \(\mathbb{Q}_2\) or something.

I think we still need to understand the operation in question in terms of the power series. 

Using the exponential \(\exp_p(z)=\sum\limits_{n=0}^\infty\frac{z^n}{n!}\) and logarithmic functions \(\log_p(1+x)=\sum\limits_{n=1}^\infty \frac{(-1)^{n+1}x^n}{n}\) defined on \(\mathbb{C}_p\), We can obtain binary functions \(a^b\) and \(log_a(b)\) that are valid for any p-adic. 

but I have absolutely no idea about binary functions \(sexp(b,h),\ slog(b,z),\ ssqrt(h,z)\) that are valid for any p-adic!