Schwartz Lemma $f(0)=0$ - meaning on $Ord_{z_0}[f(z)]$, order $1,2,3,...$?

Question

I have a question, which I want to understand for once and for all.
Let us say we have:
$f:Ball_1\left(0\right)\rightarrow \bar{\:Ball_1\left(0\right)}$
$f(0)=0$, f is analytic of course.
since I dont know anything regarding $f'(0)$, can I say that according to ord sentence, there exist:
$g\left(z\right)\in Ball_1\left(0\right),\:st:\:f\left(z\right)=g\left(z\right)\cdot z$, $g\left(0\right)\ne 0$
Or I say it like that and then it is okay:
Exist $g\left(z\right)\in Ball_1\left(0\right),\:st:\:f\left(z\right)=g\left(z\right)\cdot z$, without the not equal to 0, since we dont know the ord.
which means, can I also do in that very same meaning:
Exist $g\left(z\right)\in Ball_1\left(0\right),\:st:\:f\left(z\right)=g\left(z\right)\cdot z^2$.
And even do $z^3$, $z^4$ and such?
I hope I managed to explain clearly my trouble in this case