Unfortunately, it is in italian and not easy to find: Su certi spazi localmente convessi importanti per le applicazione, Rend. Mat. Univ. Roma (5) 14 (1955) 358-410. But as stated in my answer, you can find the parts relevant to your question in Köthe's treatise.

Well, I genuinely thought that I was providing useful information but it's your question so sorry. At the risk of being out of order yet again, I would suggest that if you are interested in subtle notions of differentiability between continous differentiability and the existence of pointwise partial derivatives, you might want to consult the specialists in real analysis (the Bruckners, David Preiss, the czech school---Fabián, Tišer) who have written voluminously on the subtelties of differentiation of functions on finite and even on infinite dimensional spaces.

This question has been answered but the following remark might be of interest since it gives a unified method which allows one to answer many related ones. Suppose that one has a topological property which is stable under the formation of closed subsets as is that of being a $k$-space (the standard name for the property in question). Then if you can find a completely regular space which fails the property, you can find a lcs which fails. This follows immediately from the fact every such space embeds in a canonical manner as a closed subspace of its so-called free lcs.

Sorry, I was being obtuse. I take your point that my additional solutions cannot satisfy your condition on the support.

Sorry, I meant the function which takes on the values $\exp x$ when $x$ and $y$ are non negative, $0$ elsewhere.