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When $F$ is local non-Archimedean, every finite extension $K$ of $F$ of degree $n$ could be embedded into your division algebra $A$. This is contained in Remark IV.4.4(c) of Milne's notes linked to in Auguste's answer.
Proposition 2.3.9(iv) in Kashiwara-Schapira's 'Sheaves on Manifolds' suggests that you might need extra conditions on your space $X$ for this to be true. Perhaps $X$ compact Hausdorff will suffice.
