Let $\tau>0$ and $E$ be a $\mathbb R$-Banach space. If $f:[0,\tau]\times E\to E$, write $\left.f\right|_1$ for the function $$[0,\tau]\to E^E\;,\;\;\;t\mapsto f(t,\;\cdot\;)\tag1$$ and $\left.f\right|_2$ for the function $$E\to E^{[0,\:\tau]}\;,\;\;\;x\mapsto f(\;\cdot\;,x)\tag2.$$ Let $\Theta\subseteq C^{0,\:1}(E,E)$ be a $\mathbb R$-Banach space and $v:[0,\tau]\times E\to E$ with $\left.v\right|_2\in C^{0,\:1}(E,C^0([0,\tau],E))$ and $\left.v\right|_1\in C^0([0,\tau],\Theta)$.
Assume $\Theta$ is continuously embedded into $C^1(E,E)$. Are we able to show that ${\rm D}_2v$ is (jointly) continuous?
Assume for the moment that $\Theta$ is continuously embedded into $C^1_{\color{red}b}(E,E)$. Then, $$\left\|f\right\|_{C^1(E,\:E)}\le c\left\|f\right\|_{\Theta}\;\;\;\text{for all }f\in\Theta\tag3$$ for some $c>0$. Now let $(s,x)\in E$ and $\varepsilon>0$. Since ${\rm D}v(s,\;\cdot\;)$ is continuous at $x$, there is a $\delta_1>0$ with $$\left\|{\rm D}_2v(s,x)-{\rm D}v(s,y)\right\|_{\mathfrak L(E)}<\frac\varepsilon2\;\;\;\text{for all }y\in B_{\delta_1}(x)\tag4.$$ And since $\left.v\right|_1$ is continuous at $s$, there is a $\delta_2>0$ with $$\left\|\left.v\right|_1(s)-\left.v\right|_1(t)\right\|_\Theta<\frac\varepsilon{2c}\;\;\;\text{for all }t\in[0,\tau]\text{ with }|s-t|<\delta_2\tag5.$$ Thus, $$\left\|{\rm D}\left(\left.v\right|_1(s)-\left.v\right|_1(t)\right)\right\|_\infty\le\left\|\left.v\right|_1(s)-\left.v\right|_1(t)\right\|_{C^1(E,\:E)}<\frac\varepsilon2\tag6$$ for all $t\in[0,\tau]$ with $|s-t|<\delta_2$. So, we can conclude that \begin{equation}\begin{split}&\left\|{\rm D}_2v(s,x)-{\rm D}_2v(t,y)\right\|_{\mathfrak L(E)}\\&\;\;\;\;\;\;\;\;\;\;\;\;\left\|{\rm D}_2v(s,x)-{\rm D}_2v(s,y)\right\|_{\mathfrak L(E)}+\left\|{\rm D}_2v(s,y)-{\rm D}_2v(t,y)\right\|_{\mathfrak L(E)}<\varepsilon\end{split}\tag7\end{equation}for all $(t,y)\in[0,\tau]\times E$ with $\max(|s-t|,\left\|x-y\right\|_E)<\min(\delta_1,\delta_2)$.
Since we should be able to restrict our considerations to a compact neighborhood of $x$, I guess we can show that the desired claim holds in the general case, where $\Theta$ is continuously embedded into $C^1(E,E)$ instead of $C^1_b(E,E)$, as well. How do we need to argue?