Of course, Muslims are allowed to swim and to be swimmers. They are allowed and encouraged to participate and practice all kinds of sports. It is forbidden neither in Quran nor by prophet Muhammad (PBUH) teachings.
However:
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Yes muslim girls and women can swim for sure; if they want; either:
Yes, they are allowed but subject to conditions.
First, the Muslim woman should wear a licit swimming suit that is covering whole body including hair and head and neck and with a fashion that is not describing or showing the woman body details or figure. There are many shops and stores that offer these Islamic swimear. The goal of the "Islamic swimwear" industry is to help and encourage more Muslim women to become physically active. Many women refrain from swimming or other water activities because of the lack of clothing alternatives. Swimming in street clothing is unsafe and unsanitary, and in most places not allowed for these reasons. Unless the pool is private for women only, Muslim women who observe hijab will not wear any type of swimsuit that does not cover the body fully. These "Islamic" swimsuit companies open up a door for such women to participate. Full-coverage swimsuits offer a compromise which is acceptable to many. If it means that more women become physically active in outdoor water sports, it can only be for the best!
Second, if the swimming pool is common for males and females, then Muslim women should observe to swim at reasonable separation spaces; away from foreign men or men who are not 'Maharem' (i.e. away from men who are entitled to marry them or subject to practice sex with them)
Yes they can but they have to wear some sort of special suit which i think is called a burkini!