Can I set up HomePod mini without a password?

See Set up HomePod - Apple Support. I don't believe the setup process requires a password as such, but you need an iOS device connected to a password protected Wi-Fi network for which your device already has the login details. See Allow others to control audio on HomePod to control AirPlay availability and whether or not that requires a password. I would generally recommend using the Anyone on Same Network option without a password.

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