From one of my club tournament rules page (likely the same for you):
"In the event of a tie of a stroke play contest, the better scoring entrant between or among the tied entrants will be the entrant with the lowest combined score for the final nine (9) holes of the Game. If any ties remain, the better scoring entrant between or among the remaining tied entrants will be the entrant with the lowest combined score for the final six (6) holes for such Game. If any ties remain, the better scoring entrant between or among the remaining tied entrants will be the entrant with the lowest combined score for the final three (3) holes for such Game. If any ties remain, the better scoring entrant between or among the remaining tied entrants will be the entrant with the lowest score for the 18th hole (9th for nine hole games) in such Game, and any remaining ties will be determined by identifying the lowest scoring entrant for each individual game hole, in sequence from 17th (8th for nine hole games) through to 1st, as is necessary to determine a winner, and if any ties remain at that point, the earlier-submitted entry between or among the tied entries will be deemed the better scoring entry.
If multiple people tie for a place other than first for a stroke play contest, the percentages for those places will be added up and divided by the number of people who tied for the place. For example, if three people tie for 2nd place, the percentages for 2nd, 3rd and 4th place will be totaled up and divided by three. Each player would receive the same amount. 5th place will then get the correct 5th place percentage."
So in short, no, there is no playoff.