If you take an infinite simple group $G$ (say Thompson’s F) and put the discrete topology on it this will have your property when k is a finite field (since the image in $GL_n(k)$ will necessarily be trivial.
Say further that $G$ is finitely presented (as indeed for example Thompson's F is). Then you can generalize this to $\mathbb{C}$ as follows by following the techniques in this paper. Basically, a non-trivial homomorphism into $GL_n(\mathbb{C})$ can be turned into a homomorphism into $GL_n(\mathbb{F}_p)$ for some prime $p$. See theorem 3.4. There they use it to turn a $\mathbb{C}$ representation with non-commutative image into a $\mathbb{F}_p$ presentation with non-commutative image, but you can do the same thing for "non-trivial" instead of "non-commutative" $\mathbb{C}$ representation. I will spell it out when I have a bit more time.
With the theorems as stated in the above paper it seems to require we assume GRH, but I don't think that's required if we don't care about a bound on $p$.