What piqued my interest in this topic is that awhile ago I went to a Hokusai exhibit, and they featured extensive works not only by himself but his daughter, who became a successful artist in her own right in 19th century Japan.
I then saw "Portrait of a Lady on Fire," in which the artist protagonist is also described as the daughter of an artist, who was trained by her well regarded and successful father. So that is the context I am thinking of specifically; if a well known / respected artist in 18th century Western Europe trained a talented daughter in his craft and then supported her having her own career, how likely would it be that she would actually be accepted and respected by society in such a role?
If the answer is yes, this is possible, would this be something that would not have been seen prior to the 18th century, or were there any earlier instances of respected professional female artists?
I've not seen the film, so I can't speak to any of its specifics, but Élisabeth Vigée Le Brun [1755-1842] would be the name I'd think of immediately.
Her father, Louis Vigée, was a portrait painter of moderate talent, not a huge success but at a time with no photography there was a steady business in painting portraits. His daughter, Elisabeth's, relationship to her father's legacy was complex, as he died when she was twelve years old and her mother remarried a wealthy jeweler who Elisabeth resented (we know this because we have her memoir and she says so).
So Elisabeth wasn't trained by her father beyond childhood, yet she did in fact become the most successful portraitist of her day. She had a talent that was appreciated by other artists and was taught by a number of artists who had been her father's colleagues. She begins a commercial painting business, but her studio is seized as she has no license; she applies to join the Académie de Saint-Luc -- which is effectively the painter's guild, and of which her father had been a member-- and is admitted.
Marie-Antoinette is a particularly avid customer -- there are perhaps thirty canvases commissioned by her from Vigée-Lebrun, portraits of herself and her family. She enjoys wealth and success, is forced to flee the Revolution, travels Europe painting the noble and wealthy -- most often their wives and daughters-- in Italy, Austria and Russia, returning to France in 1802.
Are there are other women artists with a similar career earlier? I can't think of a similarly dramatic example as close to your question, but there plenty of women artists in earlier times-- the National Museum of Women in the Arts just had an exhibition of Dutch 17th century painters, today little known figures like Judith Leyster and Magdalena van de Passe; the latter is from a family of artists too.
Sources:
Memoirs of Madame Vigée Lebrun
Woodward, Servarnne. “Les Souvenirs De Vigée-Lebrun.” Dalhousie French Studies, vol. 47, 1999, pp. 75–85.
Mary D. Sheriff. "The Exceptional Woman: Elisabeth Vigee-Lebrun and the Cultural Politics of Art". University of Chicago Press:1996
Yes, certainly! There were a number of respected eighteenth century female artists.
The most famous is Elisabeth Vigée Le Brun (1755-1842), the daughter of significantly less famous artist Louis Vigée, who died when she was little; she married a (male) painter as well. Relatively early in her career she became a portraitist to the nobility, and by the 1780s she was painting Marie Antoinette herself and a member of the Royal Academy of Painting and Sculpting. She fled the French Revolution and took up posts in other royal courts around Europe to support herself while keeping her stock high.
Angelica Kauffman (1741-1807) is another very big name in the eighteenth century art world. She was Swiss, and the daughter of a painter who trained her while they traveled around Europe, leading to her career starting at an early age. She was a member of multiple academies in different regions as well, from Florence to England.
But there are far too many for me to summarize them all: here is Wikipedia's category page for "eighteenth century women artists" on the site.
The question of how people could hold particular beliefs about women's inferiority to men while also having women with power or standing in their society comes up relatively frequently here. The fact is that gender is infinitely more complicated than a simple oppressed/oppressing dichotomy; rather than being completely marginalized by the men who dominated the art world of the eighteenth century, they found ways to make themselves part of the structure as well, though said men would rarely consider them as good as the best male artists. Several of the chapters in Women and Material Culture, 1660–1830 (Palgrave Macmillan, 2007) speak directly to the issue, making the point that women's participation in the arts went along with perceptions of what could be classified as "women's work".
Watercolors, miniatures, and wax- or cameo-carving, for instance, were easy for people to accept as fields for women because they were seen as delicate, requiring a graceful hand - similar to the way that once women began taking up positions in offices, using a typewriter was reclassified as something women would be inherently suited to as it was similar to playing the piano, a common feminine accomplishment. (Really!) People of the eighteenth century also found it relatively easy to accept "sculptresses" who did the creative work modeled in clay and had a man carve it in marble (though they were open to criticism of allowing those male assistants more creative license, and therefore the notion that they weren't really the creators of the works), and were more suspicious or scornful of women like Anne Damer (1748-1828) who wielded a hammer and chisel themselves, from remarks about the quality of their work to caricatures portraying them as too masculine.
But, that being said, they were still allowed to do their work, and were still commissioned and paid to do it, because their art was valuable and wanted. The average person doesn't not know any female artists from before 1850 because there weren't any, but because they were never allowed to be considered at the very top of their fields, and therefore weren't included in later retrospectives of the art canon.