Let $f \in Z_p[X]$, then for $ x, y \in Z_p$, $\exists a \in Z_p$ s.t. $f(y)=f(x)+(y-x)f'(x)+(y-x)^2a$.
Why is Taylor formula applicable to polynomial in $p$-adic integers $Z_p$? What condition does the remainder term have?
And can it be expanded further to higher order? But it seems strange if it can, since $\frac{f^{(n)}(x)}{n!}$ is not defined in $Z_p$ for $n\geq p$ as $n!$ is not invertable.
(I'm not 100% sure whether the formula is true, in fact I'm reading the following lemma in Serre's a course in arithmetic, and it suggest it's true. If anyone could proof specifically for $y=x+p^{n-k}z$ as below, then I will accept it as an answer.)
I have answered partially myself (this is enough for me).
It is suffice to prove for $f=X^m$, and this could be done by induction on $m$ easily.