It seems the question is not so idle after all, for the Fontaine-Mazur conjecture has something to say about it, as I discovered today in Neukirch-Schmidt-Wingberg (the NSW of Cam's answer). To paraphrase their (10.8.13), every representation

$$ \operatorname{Gal}(M|K)\to \operatorname{GL}_n(\mathbf{Q}_p) $$

of the maximal unramified extension $M$ of a number field $K$ has finite image, even if the degree $[M:K]$ is infinite. This is certainly a very strong restriction.

It seems the question is not so idle after all, for the Fontaine-Mazur conjecture has something to say about it, as I discovered today in Neukirch-Schmidt-Wingberg (the NSW of Cam's answer). To paraphrase their (10.8.13), every representation

$$ \operatorname{Gal}(M|K)\to \operatorname{GL}_n(\mathbf{Q}_p) $$

of the group of automorphisms of the maximal unramified extension $M$ of a number field $K$ has finite image, even if the degree $[M:K]$ is infinite. This is certainly a very strong restriction.