Show that for this family $T_t(x)$ of diffeomorphisms ${\rm D}T_t(x)$ is continuous in $t$ uniformly with respect to $x$

Question

Let $E$ be a $\mathbb R$-Banach space, $\tau>0$, $v:[0,\tau]\times E\to E$ and $T_t$ be a $C^1$-diffeomorphism from $E$ onto $E$ for $t\in[0,\tau]$ with $$T_t(x)=x+\int_0^tv(s,T_s(x))\:{\rm d}s\;\;\;\text{for all }(t,x)\in[0,\tau]\times E\tag1.$$

Can we show that $$[0,\tau]\times E\ni(t,x)\mapsto{\rm D}T_t(x)=\operatorname{id}_E+\int_0^t{\rm D}_2v(s,T_s(x)){\rm D}T_s(x)\:{\rm d}s\tag2$$ is continuous in the first argument uniformly with respect to the second?

I'm assuming that

1. $$E\ni x\mapsto([0,\tau]\ni t\mapsto v(t,x))\tag4$$ belongs to $C^{0,\:1}(E,C^0([0,\tau],E))$;
2. $v(t,\;\cdot\;)\in C^1(E,E)$ for all $t\in[0,\tau]$ and ${\rm D}_2v$ is (jointly) continuous.

Under this assumptions, we can show that $$E\to C^0([0,\tau],E)\;,\;\;\;x\mapsto([0,\tau]\ni t\mapsto T_t(x))\tag5$$ is continuously Fréchet differentiable. In particular, $$E\to C^0([0,\tau],\mathfrak L(E))\;,\;\;\;x\mapsto([0,\tau]\ni t\mapsto{\rm D}T_t(x))\tag6$$ is continuous.