Is there a mistake in the book : https://dropmefiles.com/UWLei
Hypothesis (H) (cf page 35)
Consider $g$ continuous on $(a,b)\times \mathbb{R}^n$, $(a,b)\subset \mathbb{R}$, and $x\mapsto f(t,x)$ continuously differentiable for all $t\in (a,b)$ .
At the top of page 36, the authors say that $f$ satisfies hypothesis (C) of page 14, which in particular means that $f$ is locally Lipschitz in $x$.
If the hypothesis is that the function is continuously differentiable, the derivatives will be continuous on a compact domain and therefore bounded. This implies that there is a uniform Lipschitz constant. If in your example the Lipschitz constant "depends on $t$", this would mean that the example does not satisfy their hypothesis.