Wallets like Bitcoin Core can let you set below 1 sat/vbyte as the fee rate. You can even use 0 sat as the fee on Bitcoin Core. But the SPV wallets that I have used before that makes you to customize fee will not allow you to use below 1 sat/vbyte.

This is mostly referred to as customization of fee and there are many bitcoin wallets that support such feature. Wallet like electrum, BlueWallet, sparrow and even some closed source wallets allow you to customize your fee. Although some wallet settings or let’s say nodes or servers you’re connected to might outrightly fail to broadcast your transaction because of low transaction fee and you could end up getting error messages