Line bundles vs. Cartier divisors on a non-integral scheme

An example is given in this note (it was credited to Kleiman).

From the exact sequence

$$1\to O^*\to K^*\to K^* / O^*\to 1$$

you see that, for as long as $H^1(K^*)=0$, the map from $H^0( K^*/ O^*)$ (i.e. Cartier divisors) to $H^1( O^*)$ (i.e. line bundles) is surjective.

On a Noetherian scheme without embedded primes (for example, reduced), $\mathcal K^*$ is the direct sum of several constant sheaves on the irreducible components, so it has trivial $H^1$.

So the example would have to be a scheme with embedded primes with a tricky nonconstant $K^*$ (the sheaf of nonzero divisors). I've seen it but can't remember right now. So this is just some general observations to narrow the search.