= Solution
A <Galton-Watson process> starts with one ancestor, $X_0=1$. Given generation size $X_r$, define
$$
X_{r+1}=\sum_{j=1}^{X_r}Z_{r,j},
$$
where all offspring variables are <independent> with common distribution $Z$. The empty sum is zero, so zero is absorbing. Its <extinction> <probability> is
$$
\eta=\mathbb P(\exists r:X_r=0)=\lim_{r\to\infty}\mathbb P(X_r=0).
$$
For the <probability generating function> $f(s)=\mathbb E[s^Z]$, the allowed fixed-point characterization is
$$
\boxed{\eta\text{ is the smallest solution in }[0,1]\text{ of }f(s)=s.}
$$
For binomial offspring, $f(s)=(1-p+ps)^k$, with $f(1)=1$, $f'(1)=kp$, and $f$ <convex> on $[0,1]$.
First assume $0<p<1$. If $k=0$, <extinction> occurs immediately. For $k=1$, $f(s)-s=(1-p)(1-s)>0$ for $s<1$, so the only <fixed point> is one. For $k\geq2$, <strict convexity> gives
$$
f(s)>f(1)+f'(1)(s-1)=1+kp(s-1)\qquad(0\leq s<1).
$$
If $kp\leq1$, the right-hand side is at least $s$, so no <fixed point> below one exists and $\eta=1$. If $kp>1$, $f(s)-s<0$ for $s$ just below one, whereas $f(0)>0$. <Continuity> supplies a <fixed point> in $(0,1)$; hence $\eta<1$. Thus in the nondegenerate parameter range,
$$
\boxed{0<p<1:\quad\eta<1\iff kp>1.}
$$
For completeness, if binomial endpoint parameters are allowed, $p=0$ or $k=0$ gives $\eta=1$, while $p=1$ and $k\geq1$ gives deterministic positive offspring and $\eta=0$. In particular $(k,p)=(1,1)$ survives forever despite <mean> one. Over $k\in\{0,1,\ldots\}$ and $p\in[0,1]$, the full condition is
$$
\boxed{\eta<1\iff kp>1\text{ or }(k,p)=(1,1).}
$$
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