Let $k$ be a characteristic zero field and consider the category $(\mathsf{Sch}/k)_\text{fppf}$ of schemes over $k$ with the fppf topology. I know that $\underline{\operatorname{Hom}}(\mathbb{G}_a,\mathbb{G}_m)=\widehat{\mathbb{G}}_a$ and so, by Cartier duality, $\underline{\operatorname{Hom}}(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=\mathbb{G}_a$. (Here we see all the groups as abelian sheaves on $(\mathsf{Sch}/k)_\text{fppf}$.)

**Is it true that the Ext sheaves $\underline{\operatorname{Ext}}^i(\mathbb{G}_a,\mathbb{G}_m)$ and $\underline{\operatorname{Ext}}^i(\widehat{\mathbb{G}}_a,\mathbb{G}_m)$ vanish for $i>0$?**

I know that [Br] shows that the Ext *groups* $\operatorname{Ext}^i(\mathbb{G}_a,\mathbb{G}_m)$ vanish for $i>0$. But I don't know how to deduce what I want from this. (This is very close to the problem in <https://mathoverflow.net/questions/247161/vanishing-of-textext2-sheaf-from-abelian-variety-to-multiplicative-group>.)

Also, I know that [BB, Lemma A.4.6] proves that $\underline{\operatorname{Ext}}^1(\widehat{\mathbb{G}}_a,\mathbb{G}_m)=0$ but I don't know how to deal with higher extensions.

References:
 - [Br] L. Breen - [Extensions of Abelian Sheaves and Eilenberg–Maclane Algebras](https://doi.org/10.1007/BF01389887)
 - [BB] L. Barbieri-Viale, A. Bertapelle - [ Sharp de Rham Realization](https://doi.org/10.1016/j.aim.2009.06.003)