Set $q=p^r$ for the finite field $\Bbb F_q$. In the formal power series ring $\Bbb F_q[[x]]$, there is a notion of convergence given by the underlying $(x)$-adic topology. If $\alpha=\sum_{n\ge0}a_np^n$ is a $p$-adic integer and we let $S_n=a_0+\cdots+a_np^n$ denote partial sums, then $f(x)^{S_n}$ should converge if $f(x)$ is contained in the translated ideal $L:=1+(x)$ (i.e. $x\mid(f-1)$). We use this to define $p$-adic powers on $L$.
The binomial series also allows us to define fractional powers on $L$ so long as the fraction's denominators are not divisible by $p$. These fractions are precisely the elements of $U:=\Bbb Z_p\cap \Bbb Q$.
Do these two definitions of powers of $U$ on $L$ agree with each other? Is there a high-altitude reason why we might expect them to agree? (Also, the first paragraph above reminds me of something called "Witt vectors" I heard about; do they fit into the picture I described above somehow?)