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Bellagamba started skating at six years old. She achieved third place in pairs and first place in singles at the Italian Acli Artistic Skating Championships. At nine, she began modern dance training with Carla Giacani and Romina Muzzi. Three years later, she attended a workshop in Florence at the Tuscany Ballet and Opus Ballet schools, where she was noticed by Daniel Tinazzi. The next year, awarded a scholarship by the same school under Cristina Bozzolini’s direction, Bellagamba moved to Florence to continue her studies and attend dance schools. At fifteen, she left school to pursue a professional dance course but continued her studies and graduated. Meanwhile, she expanded her dance education with Fabrizio Monteverde, Daniel Tinazzi, Eugenio Scigliano, Arianna Benedetti, Mauro Astolfi, Stefania Di Cosimo, Dominique Lesdema, Silvio Oddi, Steve La Chance, and Mauro Bigonzetti. In 2002, she joined the Junior Balletto di Toscana company and later Aterballetto, performing in theaters across China, New Zealand, Mexico, the United States, South Korea, and Europe.
On 5 October 2008, Alice became a student on the talent show Amici di Maria De Filippi. By the end of the program, she was the only dancer remaining and finished fourth in the show’s eighth edition. Her debut was in the musical Io Ballo, directed by Chicco Sfondrini and Patrick Rossi Gastaldi, with choreography by Garrison Rochelle, alongside other participants from Amici di Maria De Filippi. She also took part in the Amici Tour.
Director Rossella Izzo noticed her for the new Canale 5 drama Ritmo della vita. In 2009, she starred in the film Balla con noi, directed by Cinzia Bomoll, which premiered in cinemas on 27 May 2011. On 5 June 2009, she became a testimonial for the "G. SALESI ONLUS" Foundation, lending her image to promote joy and strength of life for research, improving hospital accommodations for children and their families, and supporting women and children in distress.
In 2010, she began studying acting with Francesca Viscardi Leonetti. In June 2010, she made her stage debut in Non ci posso fare niente, based on the 1782 epistolary novel Les Liaisons dangereuses by C. de Laclos. That same year, she appeared in Anna e i 5 - la nuova serie, directed by Franco Amurri. In October, she started filming the new Channel 5 series Non smettere di sognare, directed by Roberto Burchielli, which aired from March 2011.
In 2012, Bellagamba joined the cast of the TV series Provaci ancora prof 4, playing Valentina Grassetti alongside an international cast. That year, she also portrayed Salome in the television miniseries Mary of Nazareth and appeared in the drama Un passo dal cielo 2 with Terence Hill. Additionally, she starred in the sitcom Talent High School - Il sogno di Sofia as a dancer enrolled in an academy show.
In 2013, Bellagamba led the second season of the sitcom Talent High School 2 - Il sogno di Sofia, produced by LuxVide for DeAgostini. In June, she was cast by Leonardo Pieraccioni in his holiday film A Fantastic Goings, playing the character Clelia. In October, she hosted the Grand Gala of the sixth International Short Film Festival: Corti and Cigarettes. On 19 December 2013, she debuted as a TV presenter with Marche Show! - The living room of Alice on the local broadcaster Tv Center Marche.
On 30 August, at the 71st Venice Film Festival, she was awarded "Best Emerging Actress" in the first edition of the International Award dedicated to Anna Magnani.
In 2015, she played Carlotta, the first love of Olympic athlete Pietro Mennea, in the TV miniseries Pietro Mennea - The Arrow South.
That summer, she began working as a master trainer with Madonna’s Hard Candy Fitness project, giving lessons in Rome. In September, she became a Modern/Contemporary dance instructor at the Kledi Dance school in Rome, owned by Kledi Kadiu.
On 28 April 2017, at the Tulipani Festival of Seta Nera, she presented her first short film, Last Chance, which she wrote, directed, and starred in.
On 10 May 2018, the psychological thriller Le Grida del Silenzio, in which she co-starred, was released in cinemas. On 14 March 2018, she founded the Balletto delle Marche Company, serving as Artistic Director, dancer, and choreographer. After two months of rehearsals, the company performed at Miss Italy events in Marche and Abruzzo and at major festival evenings in the Marche region.