Yes, Powergirl acquired her beloved pet cat years ago during her time in the Captain Atom led Justice League Europe back in 1990. The cat was a stray ( more yellow furred than orange at the time ) with a serious bad temper, but Powergirl adored her 'Kitty'.
In one story the cat was even kidnapped when the team was based in Britain leading to Nationwide News coverage for finding the Leagues 'mascot', this was back when DC Comics had a more fun sense of humour since it's hard to imagine them doing a story like that now.
Her Cat has featured repeatedly in the background over the years and also in the amusing 2009 Palmiotti/Gray/Conner Powergirl series. She has called him several names, the most common being 'Kitty', 'Mr Cat', and in the Palmiotti/Gray/Conner series she also named him 'Stinky'.