adding one point from the Stone-Cech compactification

Let $X$ be any non-compact Tychonoff space and $beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $beta X setminus X$ is not a $G_{delta}$-set in $beta X$. Consider a topological space $X^{ast}$ which is the space $X cup {p}$
equipped with the topology induced from $beta X$. It is easy to see that if $X$ is a $sigma$-compact space, i.e. $X$ is a countable union of compact subspaces, then $p$ is $G_{delta}$ in $X^{ast}$.
I ask the following question: is the converse true?

In other words a general question is: does there exist a non- $sigma$-compact Tychonoff
space $X$ such that ${p}$ is a $G_{delta}$-set in $X cup {p}$ for any point $p$
from the reminder $beta X setminus X$?