Possible Clues:

• Counterfoil - cigarette end
• Extinguish the cigarette on a counterfoil
• Smoking tip (only half difficult!)
• Returned part of a ticket
• Pencil/cigarette remnant
• Theatergoer's memento
• Put out when objections are raised
• Counterfoil seen in the ash-tray
• Put out in small boat
• Tooth remnant surgeon finally removed from animal painter