Can we have a Non-Gaussian Likelihood in Fisher's formalism and which are the conditions or examples?

I am working on Fisher formalism and MCMC method.

First question) It seems that Fisher formalism assumes that posterior is always Gaussian : is it true ?

So if I find with MCMC a gaussian posterior, I validate the results of Fisher computation (by getting a Gaussian distribution for both).

Now, I would like to know if the Gaussianity for Likelihood is always guaranteed, independently from the PDF I use.

More concretly, if I have the likelihood with a PDF $f(x)$ :

$$mathcal{L} = big(prod_{k},f(x_{k})big)quad(1)$$

1) Under which conditions we have the Gaussianity ?

Maybe, one has to take the $log$ of $mathcal{L}$ and then write :

$$log,mathcal{L} = sum_{k},log,f(x_{k})quad(2)$$

So we could conclude that from "Central Limit Theorem", the sum on $k$ of "$log f(x_{k})$" follows, with a near factor of $N$ values $x_{k}$, a Gaussian distribution. (Actually, I think the average of random variables following the same PDF has a Gaussian $mathcal{N}(0,1)$ distribution.

Is it systematically the case for equation $(2)$ ? I mean, this is the $log$ which implies the sum and then the gaussianity with the "Central Limit Theorem" ?

So, this would be not the $mathcal{L}$ which is gaussian (in equation $(1)$) but rather the $log,mathcal{L}$ (in equation(2)).

From a rigorous point of view, I should write $f(x_k,Theta)$ instead of $f(x_k)$ alone since $Theta$ are the parameters of the model, shouldn’t I ?

3) And Finally, if the gaussianity of $log,mathcal{L}$ (or $mathcal{L}$, I don’t know for instant) is not always guaranteed, could anyone give me an example or an illustration where there is no gaussianity.

I hope to have been clear enough, feel free to ask me more precisions if need it.

Regards

ps: maybe this post should be moved to stats exchange forums.