I have such an example where the result is false for one point of the circle of convergence (cf. http://robert.rolland.acrypta.com/telechargements/exercices/OnAbelAngularTheorem.pdf).
I search an example where the conclusion is false for any point of an arc (non reduced to one point) of the circle of convergence.
A good question, I did not think about it. There are examples when the "not angular limit theorem" is false on a dense subset of the circle. This happens for functions whose radial limit is almost everywhere zero on the circle: see Lohwater's paper
http://www.ams.org/journals/proc/1955-006-01/S0002-9939-1955-0068625-6/S0002-9939-1955-0068625-6.pdf
and references in it. But I don't know if EVERY point of the circle can be a point of discontinuity for a function that has radial limits EVERYWHERE.
For sure, if such an example exists, the restriction of the function on the unit circle must have many points of continuity by the following reason. Let f be our function in the unit disc D such that $f(e^{i\phi}) = \lim_{t \to 1}f(te^{i\phi}) for every $\phi \in [0, 2\pi]$. Then $f(e^{i\phi})$ as a function in variable $\phi$ is a pointwise limit of a sequence of continuous functions, that is it belongs to the first Baire class. In such a case there is a dense $G_\delta$ subset of $ [0, 2\pi]$ at points of which $f(e^{i\phi})$ is continuous in variable $\phi$.
But the continuity of the restriction does not imply the continuity of the original function ...
The question looks very "classical" so there is a big chance that it is already answered by somebody a hundred years ago ...
Thanks for your valuable remarks. I think too that such an example is known, I will search in old classical books.
I think this dissertation:
https://core.ac.uk/download/pdf/4274093.pdf
and this page:
https://www.encyclopediaofmath.org/index.php/Boundary_properties_of_analytic_functions
and the references therein might be of interest to you.
On Omran Kouba's answer : These articles are very interesting, but as far as I could see, they do not solve the question asked.
The closed result I found is in an article by Waclaw Sierpinski (in french):
"Sur une série potentielle qui, étant convergente en tout point de son cercle de convergence, représente sur ce cercle une fonction discontinue" Rendiconti del Circolo Matematico di Palermo (1884-1940) December 1916, Volume 41, Issue 1, pp 187–190. In this paper there is an example of power series that converges everywhere on the circle of convergence (of radius 1) but which is not continuous at the point 1. This example gives a function on the closed disk which is not continuous at one point and moreover the function defined on the circle is not continuous at this point (see the remark of Vladimir Kadets on the functions defined on the circle). Then a first question could be the following: is there an example of power series converging everywhere on the closed unit disk , which is continuous on the unit circle at the point 1 and discontinuous at this point on the unit closed disk?
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