There is a paper on this topic in pages 1-7 of the September 2021 issue of The Mathematical Intelligencer. The authors are Danuta Ciesielska and Krzystof Ciesielski. If I understand correctly, the aim in their paper is to set the record straight regarding the (infamous) story about the way in which S. Banach obtained his Ph. D.
I am going to share with you the main paragraphs of the Ciesielska - Ciesielski paper below: both the phrases in boldface and the sics are mine.
*** THE STORY ***
"The story goes that Banach could not be bothered with writing a thesis, since he was interested in solving problems not necessarily connected to a possible doctoral dissertation. After some time, the university authorities became impatient. It is said that another university assistant (instructed by Stanisław Ruziewicz) wrote down Banach's theorems and proofs, and those notes were accepted as a superb dissertation. However, an exam was also required, and Banach was unwilling to take it. So one day, Banach was accosted in the corridor by a colleague, who asked him to join him in a meeting with some mathematicians who were visiting the university in order to clarify certain details, since Banach would certainly be able to answer their questions. Banach agreed and eagerly answered the questions, not realizing that he was being examined by a special commission that had arrived from Warsaw for just this purpose. In some sources [11, 19, 20], this event is described only as a possible version of events. Nevertheless, in several (mainly Polish-language) books, it is presented as a fact. There is even a book on the phobias and fears of great Poles that devotes a whole chapter to Banach and this story, claiming to demonstrate that Banach was unable to deal with his own psyche and phobias, although even this story presents Banach simply as someone who did not consider the PhD a very important acquisition."
*** DEBUNKING THE STORY ***
"... good stories aside, the truth about Banach's exam should be known. Nowadays, it is possible to check the facts, since many sources have become more easily available than they were some decades ago. It is enough to look carefully at some dates and university rules to see that the proposed account could not be accurate. Banach moved to Lvov in 1920 to take up his job at the Lvov Polytechnic. On June 24 of that year, he presented his doctoral dissertation to the Philosophy Faculty of Jan Kazimierz University. The time interval of just a couple of months was definitely too short for the university authorities to have become impatient, let alone for someone else to have written a thesis on the basis of Banach's overheard comments. Moreover, in 1920, Banach had already published three research papers. Why would he be reluctant to write a doctoral dissertation, which would be a requirement for him to keep the job?
Now let's have a closer look at the exam. According to the university rules, a PhD dissertation had to be refereed and accepted, and then two exams--in the candidate's main scientific disciplines (in Banach's case they were mathematics and physics) and in pure philosophy--were to be taken by the candidate. It turns out that the records of Banach's PhD exams have survived (they are reproduced in  and ), and we may read that Banach passed his PhD examinations in mathematics and physics. The examining board consisted of four scientists: the dean of the faculty, Zygmunt Weyberg, wo was a mineralogist; two mathematicians, Eustachy Żyliński and Hugo Steinhaus; and a physicist, Stanisław Loria. None of them was from Warsaw, and Banach knew all of them.
There is another interesting story [sic] concerning Banach's doctoral dissertation. The referees were Żyliński and Steinhaus. In October 1920, Steinhaus, who was mentoring Banach, wrote to the dean to inquire about the date of Banach's doctoral exam, for it had been four months since Banach had delivered his dissertation. The dean replied that everything was ready for the exam, but they were awaiting the referee's report (one of whom was Steinhaus himself!). Indeed, when the joint report from Steinhaus and Żyliński arrived, the exam took place immediately. Banach had submitted his dissertation on June 24, the report is dated October 30, and the exam in mathematics and physics took place on November 3. Bearing in mind that in 1920, October 30 fell on a Saturday, November 3 was therefore a Wednesday, and November 1 (Monday) is a public holiday in Poland, everything must indeed have been prepared for the exam. Banach passed this exam with a unanimous grade of 'excellent' from all four examiners.
On December 11, 1920, Banach passed the exam in philosophy (the examining board consisted of the two philosophers Kazimierz Twardowski and Mścisław Wartenberg and the dean, Zygmunt Weyberg). Banach had now fulfilled all the requirements for being granted the PhD degree, and in many sources (including a CV signed by Banach; see ), 1920 is given as the year of Banach's doctorate. However, the precise rules for obtaining a PhD from Austro-Hungarian times had been retained by Poland after regaining its independence (see ). According to those rules, the candidate was allowed to call himself a 'doctor' only after the doctoral conferment ceremony, which in the case of Banach took place on January 22, 1921. The official documents state that the academician who conferred the degree on Banach was Kazimierz Twardowski. To a mathematician, that is surprising news indeed. Why Twardowski, who was an eminent Polish philosopher? What was his connection to Banach? Could he have been his dissertation advisor? According to the rules then in force, the conferment of a new doctorate had to be celebrated by a professor from the faculty appointed by the dean, and so there is no reason to regard Twardowski as the supervisor of Banach's thesis. By analogy, one might incorrectly claim that Steinhaus's supervisor in Göttingen in 1911 was the German botanist Gustav Albert Peter, who played the same role as Twardowski in Banach's case (for details, see ).
It is frequently said that Banach was not a university graduate, so the fact that he obtained a position at the Polytechnic and a university doctorate was exceptional. This is also slightly misleading. According to the rules that were then in effect in Poland , four years of study at the university was enough for one to be eligible for a PhD, but even that requirement could be relaxed. The professors of a faculty could, at their discretion, allow someone with outstanding achievements to apply for a PhD. Moreover, in those years, there was no precise definition of who counted as a university graduate. Banach had studied at the Lvov Polytechnic for precisely four years, which was enough."
*** A KERNEL OF TRUTH? ***
"Let us dig further in an attempt to discover [a kernel of truth
underneath the gossip about Banach's doctorate].
This is a good place to recall the illustrious figure of Andrzej Turowicz (1904-1989), a mathematician, priest, and monk active mostly in Kraków, but who also spent some time working in Lvov... Turowicz knew many excellent stories, abounding in colorful detail, about mathematics and mathematicians of his time. It was not unusual for participants in various meetings that he attended to ask him to share some of his anecdotes. Whenever Turowicz had himself been a witness of an event, he recounted it with great accuracy, and one could be sure that things had really happened that way, but there were also stories he had heard from others.
On November 17, 1984, the Jagiellonian University Students' Mathematics Society (see ) invited several mathematicians to share their memories during a special meeting. Their reminiscences were taped. Turowicz was one of the guests. He contributed the anecdote about Banach's PhD exam, beginning with the words: 'This is a story I heard from Nikodym, and I am repeating it here at Nikodym's responsibility'. Turowicz recounted this event on several occasions and always credited it to Nikodym. The same attribution is also given in .
It was Nikodym whose conversation with Banach was accidentally overheard by Steinhaus in Kraków. Later, Nikodym became a prominent mathematician; after World War II he emigrated to the United States...
And it turns out that it was Nikodym who was reluctant to obtain a PhD. He used to ask: 'Will it make me any wiser?' In 1924, Nikodym (aged 35), still without a PhD, and his wife Stanisława (who was also a mathematician) moved from Kraków to Warsaw. Walerian Piotrowski made a very solid investigation concerning PhDs in mathematics at Warsaw University in the interwar period (see [24, 25]). According to , Wacław Sierpiński decided to take the matter of Nikodym's PhD exam into his own hands. He invited Nikodym to a café and began to talk with him. After a while, the dean of the department 'accidentally' appeared in the café and joined the conversation, which quickly drifted toward mathematics. More than an hour later, Sierpiński said to Nikodym: 'Congratulations. You have just passed your PhD exam.'
In our opinion, this is the source of the urban legend about Banach's doctorate. We will never know whether Nikodym gave Turowicz a twisted account of his own PhD exam, changing the main protagonist's name in the process, or whether Turowicz missed something. Our view is that the first explanation is more likely."
These are the references to which D. Ciesielska and K. Ciesielski alluded to in those paragraphs:
 D. Ciesielska, L. Maligranda, and J. Zwierzyńska. Doktoraty Polaków w Getyndze. Matematyka. Analecta 28:2 (2019), 73-116.
 K. Ciesielski. 100th anniversay of the Jagiellonian University Students' Mathematics Society. Math. Intelligencer 17:4 (1995), 42-46.
 K. Ciesielski. Lost legends of Lvov 2: Banach's grave. Math. Intelligencer 10:1 (1988), 50-51.
 T. Czeżowski (editor). Zbiór ustaw i rozporządzeń o studiach uniwersyteckich oraz innych przepisów ważnych dla studentów uniwersytetu, ze szczególnym uwzględnieniem Uniwersytetu Stefana Batorego w Wilnie. Wilno, 1926.
 E. Jakimowicz and A. Miranowicz (editors). Stefan Banach. Remarkable Life, Brilliant Mathematics. Gdańsk University Press, 2010.
 R. Kałuża. Through a Reporter's Eyes: The life of Stefan Banach. Birkhäuser, 1996.
 L. Maligranda. 100-lecie doctoratu Stefana Banacha. To appear in Wiad. Mat. 52 (2020).
 W. Piotrowski. Doktoraty z matematyki i logiki na Uniwersytecie Warszawskim w latach 1915-1939. In Dzieje Matematyki Polskiej II, edited by W. Więsław, pp. 97-131. Instytut Matematyczny Uniwersytetu Wrocławskiego, 2013.
 W. Piotrowski. Jeszcze w sprawie biografii Ottona i Stanisławy Nikodymów. Wiad. Mat. 50 (2014), 69-74.
 J. Prytuła. Doktoraty matematyki i logiki na Uniwersytecie Jana Kazimierza we Lwowie w latach 1920-1938. In Dzieje Matematyki Polskiej, edited by W. Więsław, pp. 137-161. Instytut Matematyczny Uniwersytetu Wrocławskiego, 2012.