'unceasing activity' is a correct and usable phrase in written English.
You can use it to describe something that is never inactive or that continues without pause. For example, "The ants showed unceasing activity as they moved around the yard."
Just as soon as these moments materialized, the unceasing activity swallowed them whole.
Even when, after her father died, she'd clearly been sad, she'd seemed able to convert the sadness immediately into unceasing activity on behalf of others.
In 1919 he retired, but only to a life of unceasing activity in his role as the leading authority on colonial government.
Raducanu's was a life of unceasing musical activity sustained to the very end – in August he was engaged in an elaborate jazz and poetry presentation, Dialogues and Fantasies, with the leading actor Ion Caramitru.
It may be that one day fairly soon this slice of prime real estate will be occupied by a skyscraper like its companions near Central Park – if not quite as expensive or as tall – but for now the space has become a small area of solidarity and contemplation in a city better known for its individualism and unceasing, feverish activity.
In the United States, lexicographical activity has been unceasing since 1828.
Life is an unceasing process of action, reaction, and consequence.
Dr. Jacques Rogge, the I.O.C. president and a fellow Belgian, said, "His ideas and energy helped to begin an unceasing fight against doping and any activity damaging the integrity of the sport".
Florida is America's bedraggled paradise, and its most fervent activity, of course, is the reshaping of nature through remorseless and unceasing development.
It's an extended Swiss Family Robinson experience with planned activities, cheery young staff members and group meals all set to an unceasing soundtrack of tree frogs.
Their effort was unceasing.
