First note that in a space with countable tightness: $P$-point $\Leftrightarrow$ weak $P$-point $\Leftrightarrow$ Isolated point. So you are looking for a point for which every deleted neighborhood contains a non-isolated point. You can do a lot better:

Let $X$ be a countable *maximal space* (i.e. such that the topology on $X$ is maximal among those topologies which have no isolated points). This is an easy application of Zorn´s lemma, starting with any countable space which has no isolated points. Then:

1) $X$ is countable (in particular it has countable tightness),

2) No point of $X$ is a limit point of two disjoint subsets of $X$ (in particular it has no non-trivial convergent sequences and therefore it is not Fréchet),

3) $X$ has no isolated points (so **any** point has the property that you want).

Property 2 is not so easy to check, but you can look at **"Applications of maximal topologies"** by E.K. van Douwen, where he proves (see Theorem 2.2) that for *crowded* spaces (i.e. spaces with no isolated points) property 2 (which he calls *perfectly disconnected*) is equivalent to being a maximal space. In the same paper, van Douwen shows how you can construct such a space which is also regular (this is not a simple application of Zorn´s lemma anymore).

Descriptive Topology in Selected Topics of Functional Analysis, Developments in Mathematics, Vol. 24, Springer Science+Business Media, New York 2011. You will find there a large supply of examples you seek (mostly in the realm of function spaces), and it is very likely that the answer to your question can be deduced from the results gathered in this book. – Tomek Kania Aug 21 '13 at 13:11