Trouble broke out when hundreds of protesters gathered near the Democracy Monument, a focal point of three months of demonstrations to demand a new constitution and the removal of Prime Minister Prayuth Chan-ocha, a former junta leader.
Protesters pushed up against a police line and some threw blue paint over the officers. Police meanwhile demolished a tent that had been set up by the protesters.
A Thai police spokesman said some people for whom there were outstanding arrest warrants had been detained and that they would now face the legal process. There had been "four or five" arrests, he said.
Thai Lawyers for Human Rights said at least nine people had been arrested and that one was injured.
Protests since mid-July have brought the highest pressure in years on Thailand's ruling establishment, with protesters also calling for reforms to curb the monarchy of King Maha Vajiralongkorn, once a taboo subject.
Tuesday was a public holiday to mark four years since the death of the king's father, King Bhumibol Adulyadej.