a faucet that can give even 1 BTC satoshi immediately?

A faucet can not pay 1 satoshi because it falls below the dust limit used by the majority of network nodes. In addition, the fees required to send a transaction would total well over 1 satoshi. Even if it was possible to confirm such a transaction, it would cost more than the amount transferred.

If you have a Lightning network node with the appropriate inbound capability, you can accept a Lightning Network payment on 1 satoshi. It should be noted that if the channel was closed with only 1 seated on your side, you would not be able to receive chain transmissions for this amount, due to the dust limits mentioned above.

Note that service recommendations are off topic hereI will not comment on the current availability of such a faucet.