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Consider for $\delta>0$ the function $$\vec F:\mathbb{R}^3\to\mathbb{R}^3:\vec F(x,y,z)=\begin{pmatrix}-x-\delta(y^2+z^2)+1\\-y+\delta xy\\ -z+\delta xz\end{pmatrix}.$$

Is this map locally Lipschitz? (Recall that $\vec F$ is locally Lipschitz if for all $a\in\mathbb{R}^3$ and all $r>0$ there is a $K\in \mathbb{R}$ that satisfies $$|\vec F(x)-\vec F(a)|\le K\cdot|x-a| $$ for all $x$ in the open balls centered at $a$).

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Yes, every $C^1$ map is locally Lipschitz (mean value theorem).

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Well, your vector field $F \in C^{1}(\mathbb{R}^{3})$. This is a sufficient condition for it to be locally lipschitz: indeed, by Weierstrass' theorem $F^{\prime}$ (I mean the jacobian, of course) is bounded on every compact subset of $\mathbb R^{3}$ and therefore the function is locally Lipschitz (to prove this you may also use the mean value theorem for function of more variables).

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