Boundless of $\nabla_{\theta} \log p_{\theta}$ with $p_{\theta}\in C^{1,1}$

Question

let us assume to have a probability density function $p_{\theta}(x, y)$ defined on a compact set $\mathcal{X}\times \mathcal{Y}$ and with $\theta \in [0,1]^n$. Let us assume that $\theta\mapsto p_{\theta}(x,y)$ is of class $C^{1,1}$ uniformly in $(x,y)$. Is it true that $|\nabla_{\theta} \log p_{\theta}|$ is bounded?

Thanks!