From MathSciNet:

Thomas, Simon Infinite products of finite simple groups. II. J. Group Theory 2 (1999), no. 4, 401–434.

Summary: "(...) In the course of our classification proof, we also show that if $K$ is a field of cardinality $2^\omega$ and $G$ is a non-trivial linear group over $K$, then there exists a subgroup $H$ of $G$ such that $1<[G:H] \le \omega$."

[This is almost the same since enlarging $G$, we can always assume $G$ is simple, say $G=\mathrm{PSL}_m(K)$, and then the induced homomorphism $G\to \mathrm{Sym}(G/H)$ is injective. The proof uses countable valuations as well.]

From MathSciNet:

Thomas, Simon Infinite products of finite simple groups. II. J. Group Theory 2 (1999), no. 4, 401–434.

Summary: "(...) In the course of our classification proof, we also show that if $K$ is a field of cardinality $2^\omega$ and $G$ is a non-trivial linear group over $K$, then there exists a subgroup $H$ of $G$ such that $1<[G:H] \le \omega$."

[This is almost the same since enlarging $G$, we can always assume $G$ is simple, say $G=\mathrm{PSL}_m(K)$, and then the induced homomorphism $G\to \mathrm{Sym}(G/H)$ is injective. The proof uses countable valuations as well.]

Ershov, Yu. L. ; Churkin, V. A. On a problem of Ulam. (Russian) Dokl. Akad. Nauk 399 (2004), no. 3, 307-309.

Theorem 1. Every linear group over a field with continuum cardinality (in particular over the field $\mathbf{R}$ of all real numbers or over the field $\mathbf{C}$ of all complex numbers) can be embedded (as an abstract group) in the permutation group of a countable set.

(Unlike S. Thomas, they were aware of Ulam's question. On the other hand they were visibly not aware of S. Thomas paper.)