The universal central extension $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is the preimage of $\text{Sp}_{2n}\mathbb{Z}$ in the universal cover of $\text{Sp}_{2n}\mathbb{R}$, and fits into the sequence

$1\to \mathbb{Z}\to \widetilde{\text{Sp}_{2n}}\mathbb{Z}\to \text{Sp}_{2n}\mathbb{Z}\to 1$.

Deligne proved that $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is not residually finite; the intersection of all finite-index subgroups of is $2\mathbb{Z}<\widetilde{\text{Sp}_{2n}}\mathbb{Z}$. In particular, this implies that $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is not linear. But certainly $\mathbb{Z}$ and $\text{Sp}_{2n}\mathbb{Z}$ are. If you want an arithmetic group, you can take the corresponding $\mathbb{Z}/k\mathbb{Z}$-extension of $\text{Sp}_{2n}\mathbb{Z}$, which will not be linear as long as $k\neq 2$.

I learned the proof of this theorem from Dave Witte Morris, who has written up his fairly-accessible notes as "A lattice with no torsion-free subgroup of finite index (after P. Deligne)" (PDF link).

The universal central extension $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is the preimage of $\text{Sp}_{2n}\mathbb{Z}$ in the universal cover of $\text{Sp}_{2n}\mathbb{R}$, and fits into the sequence

$$1\to \mathbb{Z}\to \widetilde{\text{Sp}_{2n}}\mathbb{Z}\to \text{Sp}_{2n}\mathbb{Z}\to 1.$$

Deligne proved that $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is not residually finite; the intersection of all finite-index subgroups of is $2\mathbb{Z}<\widetilde{\text{Sp}_{2n}}\mathbb{Z}$. In particular, this implies that $\widetilde{\text{Sp}_{2n}}\mathbb{Z}$ is not linear. But certainly $\mathbb{Z}$ and $\text{Sp}_{2n}\mathbb{Z}$ are. If you want an arithmetic group, you can take the corresponding $\mathbb{Z}/k\mathbb{Z}$-extension of $\text{Sp}_{2n}\mathbb{Z}$, which will not be linear as long as $k\neq 2$.

I learned the proof of this theorem from Dave Witte Morris, who has written up his fairly-accessible notes as "A lattice with no torsion-free subgroup of finite index (after P. Deligne)" (PDF link).