Example problem that is not in $2^{o(n)}$ but could be solved in $O(2^{cn})$ for any $c > 0$ (suggested by wording of ETH)
Yes, an example is:
Accept $0^k 1^m$ iff $k&gt;0$ and the $k$th Turing machine halts in less than $2^{m/k}$ steps on the empty input.  Other strings are rejected.  See question Does $∩_{ε&gt;0} \mathrm{DTIME}(O(n^{2+ε})) = \mathrm{DTIME}(n^{2+o(1)})$? for the proof.

I expect that there is a relativization barrier against proving equivalence of ETH with its weakening using $2^{o(n)}$, and that there are oracles relative to which (note that this is not ETH) for every $ε&gt;0$ every relativized NP-complete problem can be solved in time $O(2^{n^ε})$ but not in time $2^{n^{o(1)}}$.