Is unstable system unstable for ALL inputs?
Yes. There are inputs for which the output of an unstable system is bounded.
For example, we can stabilise some unstable systems with feedback. In this case, someone who sees only the input and output of the unstable system (without seeing the feedback mechanism) would see bounded outputs for the unstable system.

However, you may not be able to find the DC gain if the DC input is not in the list of inputs for which the (unstable) system has bounded outputs.

Edit

Consider as an example, the first order system A
$$
\frac{dx}{dt} = x+u\\
x(0) = 1
$$
The state is bounded for the input \$u(t) = -1\$. The output grows for all other constant inputs.