Yes, it depends on the mint. If a mint goes offline indefinitely, you could lose your sats. That’s why it’s advised to only use trusted mints and to diversify your sats across multiple mints. Mint index: https://cashumints.space

You can also host your own mint, although this requires some server knowledge. Alternatively, you can easily create a Nostr mint on an LNbits server. https://docs.cashu.space/mints