No. Bali is not a country. It is a province of Indonesia—one of 38 provinces in the world's 4th most populous nation.
The smallest country in the world is Vatican City (0.44 km², ~800 people). The smallest Indonesian province by area is Jakarta (special capital region); by population, North Kalimantan (~700K).
Confusion arises because Bali has distinct culture, international airport, and global brand recognition—almost like a "country within a country" for tourists.