Question: Examples of this? I know this is possible with infinite fields but I'm not sure about finite fields.

Answer: 
Yes. For example the finite fields $\mathbb{F}\_4$ and $\mathbb{F}\_2$ both have characteristic 2, but they dont have the same number of elements so can't be isomorphic.

More generally, there exists a unique finite field (up to isomorphism) which has characteristic $p>0$, and $q=p^n$ elements denoted $\mathbb{F}\_q$.
Every finite field $F$ fails to be algebraically closed. One can write
$$
f(x) = 1+\prod\_{a\in F}(x-a)
$$
and it's easy to see that this polynomial has no roots in $F$. There is therefore a larger field, of which $F$ is a subfield, having roots of this equation. That larger field has the same characteristic.