No. There is no timer, daily mode or public leaderboard. Your progress and hints are saved in this browser when storage is available.
Word Search is untimed, so you can take as long as you like.
What is saved
The words you have found and the hints you used are saved in this browser, so you can leave and return to the same grid. Nothing is sent to a public leaderboard. When you finish, you can share your result, which includes the hints you used.
For ways to find words faster, read the Word Search tips and strategy.