The general verdict expressed in @Jos' answer is absolutely correct. It was not customary for young upper-middle class women to chew gum in public in German-speaking Europe.

However, in the given or intended context there might be some additional aspects to consider.

Chewing in public was seen as "eating" and that was very highly frowned upon until the American soldiers came. They were frowned upon, their behaviour was frowned upon. And part of that scandalous lack of etiquette was eating in the streets while on patrol or jsut whenever they were hungry. But this frowning was mostly restricted to some strata of the older parts of the upper and middle classes, while younger and lower classes quickly adopted this (lack of) style. (Compare the last two chapters of Gunther Hirschfelder: "Europäische Esskultur: Eine Geschichte der Ernährung von der Steinzeit bis heute", Campus: Frankfurt, 2001.)

But that was valid for both sexes. When looking especially at women, their behaviour and how it was or is judged, there might be traces of a certain symbolism found in movies, like Pretty Woman and Trading Places where both female protagonaists are constantly chewing gum to mark them as prostitutes. That might be a bit harsh and is from American movies, but then they are also quite more recent than the 1930s.

Concerning the situation in Europe it has to be added that just like after the second world war, chewing gum was largely seen as typical American from the turn of the century:

Gummikauen ist die wohlfeilste Art, sich zu amerikanisieren, darum haben die Deutschen von heute, die eine starke Amerika-Sehnsucht in sich tragen, sie eben gewählt. Das heißt: sie sind von den Herren des Kaugummis als ein prädestiniertes Volk auserwählt und erfolgreich behandelt worden. Und heute sind sie kaugummireif. Dies, daß ein Ladenhüter zum Modeartikel werden, daß eine stille kleine Sekte, die unauffällig an einer alten Gewohnheit klebte, zu einer von der Neuheit ihres Tuns überzeugten Massenbewegung anwachsen konnte, zeugt mehr als anderes für die Macht der Reklame.
Translation: Chewing gum is the cheapest way to Americanize, which is why the Germans of today, who have a strong yearning for America in them, have chosen it. In other words, they have been chosen and successfully treated by the masters of chewing gum as a predestined people. And today they're chewing gum. The fact that a shopkeeper became a fashion item, that a quiet little sect that inconspicuously stuck to an old habit could grow into a mass movement convinced of the novelty of its actions testifies more than anything to the power of advertising.
_{Ernst Lorsy, „Die Stunde des Kaugummis” (1926), cited from DGDB: Die Weimarer Republik (1918/19-1933)}

Were there ads?

_{Recent reproduction, original undated? But style depicted seems to point at a date before WW2?}

While that "typically American" is an epithet in upper classes, it also stood for modernism and progress. Those that admired America and its modernity where also emulating American trends and behaviour, including chewing gum.

Die praktischen, die smarten Amerikaner! Das Girl haben sie bei uns eingeführt und den Jazz und die Songs mit Rachenkatarrh, und die amerikanische Buchhaltung und Roberts und fast sogar auch den Kaugummi. Wissen Sie, dass jeder fünfte Amerikaner sein eigenes Auto hat? Bitte schön: Hut ab! (The practical, the smart Americans! They introduced the girl to us and the jazz and the songs with throat catarrh, and the American accounting and Roberts and almost even the chewing gum. Do you know that every fifth American has his own car? Here you go. Kudos.) Volkszeitung (1890-1904) /Berliner Volkszeitung (1904-1930), 13 September 1929.

Looking at the classic conduct books (in German often erronously called Knigge this topic is not listed).

A long list of the most important books on the topic is Literatur / Bibliography (1715-2001) – Gutes Benehmen - Good manners.

It is quite astonishing to look for early examples of expressions of opinions on chewing gum.

It was known in Europe and used in Europe, sometimes even recommended –– as a substitute for smoking.

Compare this novel by Ilse Frapan-Akunian: "Zwischen Elbe und Alster ", 1890.

„Was ißt du, Theodor?“

„Ich eß nich,“ brummt der Kleine vorwurfsvoll und nimmt ein schwarzes Klümpchen zwischen den Zähnen heraus, um es mit den Fingern hochzuhalten, „ich krieg ja all die ganse Woche nix mit, – is bloß ’n büschen Kaugummi.“

Before the Second World War, even before the first, it was already in circulation. But apart from being a novelty food, it was also an import. Meaning it was not so ubiquitously cheap as in the US, or after 1932 even hardly available.

Als der Kaugummi nach dem Zweiten Weltkrieg auch Europa eroberte, mussten noch einige Skeptiker überzeugt werden, die den Kaugummi etwas vulgär und möglicherweise schädlich für Zähne und Magen fanden. In der Werbung der 50er-Jahre hingegen war man nicht zimperlich. «In jedem Kind schlummert ein Kauer. Nägel, Gummis, Bleistifte, Bücher und Lineale sind eine leichte und gefährliche Beute für die kleinen Zähne. Schützen Sie sie. Geben Sie Ihrem Kind ein heilsames Mittel, um seine Nerven zu beruhigen und sich zu entspannen.»
Charly Veuthey: "Aufgeblasen", Cooperation (coop Zeitung), 09.07.2018

And keep in mind that your readers are interpreting this not only from an historical accuracy perpsective, but now. Chewing gum in public is still a (weak) divider. Last year I met a young East-German woman expressing exactly this sentiment from an old West German aristocrat:

Thurn und Taxis: Wenn einer Kaugummi kaut, finde ich das extrem irritierend.
_{Julika Meinert: „Sonst kommen wir so rüber wie die Russen“, Welt, 23.03.2013.}

Looking especially at woman:

American women love chewing gum, keeping this sweet something in their mouth all day long. (my translation) O Elsner: "Die Welt der Technik, Volume 74", 1912 So at least the German readers were told that this was a (stereo) typical behaviour of American women.

How does this translate into the European context at that time:

Die Reklame der Tageszeitungen preist ein neues Genuss- und Heilmittel an: „Amerikanischer Kaugummi (Chewing … bei Männern. sondern auch bei Frauen. Beliebt bei jungen Damen, bei Knaben und Mädchen, bei Schwarzen. wie bei Weissen.
(The advertising of the daily newspapers praises a new pleasure and remedy: "American chewing gum (Chewing... for men. but also for women. Popular with young ladies, boys and girls, blacks and whites.) J.F. Lehmann: "Münchener medizinische Wochenschrift, Volume 57", 1910.

Or the verdict

Miß Nellie M. Horton war vor zehn Iahren Stenographin in dem Comptoir eines Pepsin-Fabrikanten in Cleveland. Wahrscheinlich kaute sie Gummi, eine üble Gewohnheit, die bis jetzt glücklicherweise hauptsächlich auf Amerika beschränkt blieb.
Pustertaler Bote, 18 October 1901

The official introduction of chewing gum into German society is often wrongly dated. That is because:

Since the end of World War II, chewing gum has had an important place in the collective memory of Germans. It is a symbol of the new beginning after the war and a result of the U.S. occupation of parts of western Germany. Americanization of West German society during the 1950s brought Coca-Cola, chewing gum, and rock ’n’ roll to a young generation that was starving for entertainment and willing to experiment with new and unusual cultural imports. The history of chewing gum in the Federal Republic of Germany (FRG) is the history of the product developed by the Wrigley’s Company, which was founded in 1891 in Chicago. Wrigley’s established its first German production facility in 1925 in Frankfurt am Main, where it produced the PK gumball. However, in 1932 Wrigley’s had to close its German outlet because of increasing difficulties importing necessary raw materials and the financial problems related to Germany’s obligations under the Treaty of Versailles and the Great Depres- sion. After 1933, Wrigley’s chewing gum could still be distributed and purchased inside Germany. However, it was no longer produced there and had to be imported. Wrigley’s opened an import and distribution bureau at Berlin’s famous Unter den Linden Boulevard for marketing purposes. In contrast to Coca-Cola, however, Wrigley’s renamed its product: chewing gum became Kaubonbon (chewable candy).
The image of the American GI chewing Wrigley’s gum became an icon in European culture during and after World War II. For some time, Wrigley’s decided to send its entire production to the U.S. soldiers engaged in the European and Asiatic theatres of World War II. As well as Coca-Cola, chewing gum became symbolic of U.S. resolve to win the war. It was to give the U.S. soldier psychological support during combat. Scientific studies had proven that chewing helped to decrease stress. Furthermore, chewing gum became important in winning the trust of the defeated civilian population.
Thomas Adam: "Germany and the Americas. Culture, Politics, and History A Multidisciplinary Encyclopedia. Volume I", ABC-Clio: Santa Barbara, Denver, 2005, p227.

Coming back to the movie references and the practice of women chewing gum at roughly the inquired time and the advertis*m*nts, this one disguised as editorial content ad ties it all together:

Kleiner Leitfaden für Küssende:
[…] Ziehe sie sanft an dich. Beeile dich nicht. Starre in die Liebesglut, die in ihren Augen glimmt. Seufz noch einmal. Neige deinen Kopf dem ihren zu. bis deine Lippen — Doch warte! Küsse sie nicht, bis du gewiß bist, daß sie den Antiseptischen Kau-Gummi der Firma L. ?1. gebraucht, den einzigen Gummi, der das Küssen ungefährlich macht. Wenn sie ein antiseptisches Gummi-Girl ist, dann küsse sie." Neue Hamburger Zeitung 21 December 1912

Translation: Gently draw her to you. Don't hurry. Stare into the glow of love that glows in her eyes. Sigh again. Tilt your head towards hers until your lips - Yes, wait! Don't kiss her until you are sure that she uses the antiseptic chewing gum from XM, the only gum that makes kissing harmless. If she's an antiseptic rubber girl, kiss her."

Conclusion

As I already wrote on Jos' answer: it was certainly not customary for young women to chew gum in public. It was certainly very much frowned upon. But showing a young girl at the time chewing her gum makes this all the more fitting if she is supposed to be rebellious.