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The following result is due to W. Burnside:

Theorem. Let $G$ be a subgroup of $\textrm{GL}_n(\mathbb{C})$. If $G$ has finite exponent, then $G$ is finite.

The proof relies on the following:

Lemma. Let $A\in\mathcal{M}_n(\mathbb{C})$ such that for all $k\in\{1,\ldots,n\}$, $\textrm{tr}(A^k)=0$, then $A$ is nilpotent.

It is not hard to see that the theorem still holds over fields of characteristic zero. Indeed, to establish the lemma it suffices to consider a splitting field of the characteristic polynomial of $A$. However, the theorem fails to be true over infinite field of prime characteristic; consider the following subgroup of $\textrm{GL}_2(\mathbb{F}_p(t))$: $$G:=\left\{\begin{pmatrix}1&f\\0&1\end{pmatrix};f\in\mathbb{F}_p(t)\right\}.$$ Notice that $G$ is infinite albeit having exponent $p$. My conjecture is that the following refinement is true:

Conjecture. Let $k$ be an infinite field of prime characteristic $p$ and let $G$ be a subgroup of $\textrm{GL}_n(k)$. If the exponent of $G$ is finite and prime with $p$, then $G$ is finite.

I already proved the conjecture for $n

Any enlightenment and/or references will be greatly appreciated!

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    This may be nonsense, but it looks similar to the problem you get if you try and use normal characters in characteristic $p$. So perhaps you the lemma will be correct if you use the Brauer character in place of the trace. (I haven't got time to think about it right now.) – 2017-01-23

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The version of Burnside's theorem given in B.A.F. Wehrfritz' book "Infinite Linear Groups" as Corollary 1.23 states that a completely reducible subgroup of $GL(n, F)$ of finite exponent $e$ has finite order at most $e^{n^2}$.

The theorem of Maschke-Schur is phrased in the cited book in the following form (as Corollary 1.6): A locally finite subgroup $G$ of $GL(n, F)$ is completely reducible, if either char F = 0 or $G$ does not contain any element of order char F. (A group is called "locally finite" if all its finitely generated subgroups are finite.)

By a theorem of Schur (Corollary 4.9 in the same book) a periodic linear group is locally finite. (A group is called "linear" if it is a subgroup of $GL(n, F)$ for some (commutative) field F and some $n\in \mathbb N$. "Periodic" means that all elements have finite order.)

Summary: Your hunch is correct. It's a classical result in the theory of groups.