While accepting Jade's and Ruben's answers, I'd like to add something that I've read about talipot fan.
If we look at the story of seventh verse of Arahanta Vagga in the book, "The Dhammapada - Pali text & translation with stories in brief & notes" (1963) by Ven. Narada, it has mentioned that;
"While attempting to awaken, with his fan, a pupil novice who was thought to be asleep, a teacher accidentally injured the pupil's eye. Later, when the teacher tried to close the door, he injured the pupil's hand too. But the pupil showed no resentment at the carelessness of his teacher. He was well trained. The Buddha praised him"
So, the origin of talipot fan may considered to be from India during Tathagata's time.
In the book, "An Historical Relation of the Island Ceylon in the East-Indies" (1681) by Robert Knox, it has mentioned that;
"It is a yellow coat gathered together about their waist, and comes over their left shoulder, girt about with a belt of fine pack-thread. Their heads are shaved, and they go bare-headed and carry in their hands a round fan with a wooden handle, which is to keep the sun off their hands.
They have the honour of carrying the Tallipot with the broad end over their heads foremost; which none but the kings does: wheresoever they come, they have a mat and a white cloth laid over upon a stool for them to sit upon; which is also an honour used only to the king."
Nowadays in Sri Lanka when a Thero receives an honorary title, he is presented with a decorated talipot (vijinipath) and a scroll.
From the story of Goddess Laja, we can see that Venerable Mahakashyapa had used the word "vijinipath = talipot fan" without describing it. So it could have been a common instrument at that time also. From these, we can conclude that talipot fan was used from ancient time.
"Goddess, disappear l said. Erudite preachers of the future who mount pulpits with vijinipath in their hand should not be heard to say that a certain goddess came down to help Venerable Mahakashyapa with his day to day chores. Now, go away and never come again."