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I can't answer your question completely, but this extended comment may be helpful. Probably the answer will be "yes", judging at least from the old classification by Tits summarized in the proceedings of the 1965 AMS Summer Institute at Boulder (published by AMS in 1966 as vol. 9 in their series Proc. of Symposia in Pure Math., which was for a while freely available at the AMS website). Tits was interested in sorting out the possible forms of a simple algebraic group over various kinds of fields, and in his summary tables one sees case-by-case that existence of a group defined over what he calls a "$\mathfrak{p}$-adic" field implies existence of such a group over some number field (but not vice versa). He also provides some general theorems in arbitrary characteristic, with proofs later filled in and somewhat corrected in thesis work by his student Martin Selbach at Bonn (1976). But it may be impossible to find a complete treatment anywhere in the literature. (Will's comment is worth following up.)