I say a field $K$ is cool if every group homomorphism between to $K$-vector spaces is $K$-linear.
$\mathbb{Z}_p$ is cool because multiplication by a scalar is like an iterated sum.
$\mathbb{Q}$ is cool too. Lef $f$ be such a group homorphism. $$bf(\frac a b x)=f(\frac a b x)+\ldots +f(\frac a b x)=f(\frac a b x+\ldots+\frac a b x)=f(ax)$$ (the dots mean "$b$ times") and hence $$f(\frac a b x)=\frac a b f(x).$$
The field of $p$-adic rationals are cool too (same argument).
$\mathbb{F}_4=\{0,1,a,a+1=a^2\}$ is not cool. The map $$0\mapsto 0,1\mapsto 1,a\mapsto a+1,a+1\mapsto a$$ repsects sum but not multiplication by $a$.
$\mathbb{R}$ is not cool. Let $\{x_1,x_2,\ldots,x_i,\ldots\}$ be an (uncountable) basis for $\mathbb{R}$ as a $\mathbb{Q}$-vector space. The map $$(\alpha_1,\alpha_2,\ldots,\alpha_i,\ldots)\mapsto (\alpha_2,\alpha_1,\ldots,\alpha_i,\ldots)$$ (swapping the first two components) is $\mathbb{Q}$-linear and so respects sum but it's not $\mathbb{R}$-linear because $x_3\mapsto x_3$ but $x_1\mapsto x_2\neq x_1$. The algebraic neither are cool by the same argument.
I wondered if cool fields have been studied before, and under which name! What's the pattern behind them?
For me the link between (and reason for coolness of) $\mathbb{Z}_p$ and $\mathbb{Q}$ is that both are the fields of fractions of the only cyclic integral domains. Maybe coolness is hereditary and the $p$-adic rationals are hence naturally cool.
The only cool fields are $\mathbb{Q}$ and $\mathbb{Z}/p\mathbb{Z}$. So "cool field" is a synonym for "prime field".
You showed that the prime fields are cool. Now suppose $F$ is not a prime field. Let $k\subsetneq F$ be the prime subfield of $F$. Then $F$ is a $k$-vector space of dimension at least $2$, so it admits a nontrivial $k$-linear automorphism, which is an additive group automorphism, but which is not $F$-linear (since the only $F$-linear automorphism of $F$ is the identity).
In particular, the $p$-adic field $\mathbb{Q}_p$ is not cool.