Here is an example.

Let $K = \mathbf{Q}(\sqrt{-21})$. Then the class group of $K$ is $C_2 \times C_2$ and its Hilbert class field is $H_K = \mathbf{Q}(\sqrt{-1}, \sqrt{3}, \sqrt{7})$. In particular, $H_K$ is a CM-field and its maximal totally real subfield $H_K^+$ is $\mathbf{Q}(\sqrt{3}, \sqrt{7})$.

The LMFDB database has an elliptic curve [4.4.7056.1-1.1.a1][1] over $H_K^+$ which has everywhere good reduction and has CM by $\mathcal{O}_K$. Base-extending this from $H_K^+$ to $H_K$ gives the example you seek.

Similar examples should exist whenever $H_K$ has class number 1 and all units of $H_K$ are in the kernel of the norm map to $K$. (The class number condition may well not be needed, but the condition on the units certainly is). I have no idea if there are infinitely many such fields $K$, but there definitely some! This happens for $\mathbf{Q}(\sqrt{-d})$ for $d = 21, 33, 42, 57, 66, 77, 93$ (and no others for $d < 100$).

  [1]: https://beta.lmfdb.org/EllipticCurve/4.4.7056.1/1.1/a/1