Who is Kirszbraun?

This Banach biography web page in Polish says that Kirszbraun was born in 1903 or 1904 and died in 1942 (he is listed there among other Polish mathematicians who died in the course of WW II). Further googling revealed that his full name was Mojżesz David Kirszbraun. According to the Zentralblatt Math. database, he wrote just one paper in German (just as Mark mentioned in the comments):

Kirszbraun, M.D. Über die zusammenziehenden und Lipschitzschen Transformationen. (German) Fundam. Math. 22, 77-108 (1934).

Here is the link to the review of this paper by Freudental.

Perhaps some Polish colleague(s) could provide further details.

Apparently he has an entry written by Edward Marczewski in the polish biographical dictionary , according to the official site of the publisher. Somewhat suprisingly there seem to be no library which has this dictionary where I live so I can't help you any further.

Best of luck! I know the feeling - I once wanted to learn anything about Wegge-Olsen who wrote a nice book on K-theory. I haven't found anything in internet, but in the end I asked Paul Baum accidentally and he told me a bit about him.

It is very, VERY likely that Kirszbraun published only one paper. Namely, if you read carefully that short entry in Polish (written by E.Marczewski = Szpilrajn), you'll notice that in the penultimate sentence it says that upon completing his studies he got a job as an actuary ("aktuariusz", the person who calculates insurance costs) in an insurance company named "Przyszlosc"; therefore, it is quite reasonable to assume that he didn't continue any form of a(n academic) career as a research mathematician. On the other hand, a footnote an the title page of its Fundamenta paper clearly indicates that the paper is an abridged and improved version of his "Magister" (= Master) thesis, defended back in 1930 and prepared in the preceding 4-year period. So, it is rather safe to jump to the conclusion that Über die zusammenziehenden und Lipschitzschen Transformationen is his only publication, and that he didn't pursued any further mathematical research.