# Galois representation with infinite image but finite image everywhere locally

Fix a prime $$l$$. Let $$\phi:\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\to GL_n(\mathbb{Q}_l)$$ be a semisimple continuous representation. Assume $$\phi$$ has finite image when restricted to $$\mathrm{Gal}(\overline{\mathbb{Q}_p}/\mathbb{Q}_p)$$ for any $$p$$ (including $$p=l$$). Can $$\phi$$ have infinite image?

Since $$\phi$$ is de Rham at $$l$$, to have infinite image it must be ramified at infinitely many places. Examples of semisimple infinitely ramified Galois representations were constructed in a paper of Ramakrishna (note that the ramified primes necessarily have zero density).