Let k be a field. Let G be a separated group algebraic space locally of finite type over k. There does not exist a nonconstant morphism f : P^1_k → G over Spec(k). See Lemma Tag 0AEN.
Slogan: no (complete) rational curves on groups.
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Surely separated is not necessary. I seem to recall from Raynaud’s “Specialization of the Picard functor” that for every finite type group scheme, there is a universal quotient group scheme that is separated. Presumably you could deduce the non-separated case by deducing the composition of the morphism from $\mathbb{P}^1$ with this universal quotient is a constant morphism.
OK, I just added this lemma because of the example by Bhargav of a non-algebraic Hom stack (see chapter on examples) and I didn’t take the time to get the best result. But on the other hand, Raynaud doesn’t consider a group algebraic space such as A^1/Z (highly nonseparated…) and it was not immediately clear to me how to deal with those in an elementary way. Still as you say, it should be true.