If downtime is an not an option, Lambda is not the solution. Amazon will automatically shut down instances of your Lambda if they haven't been used (~5-15 minutes), and starting a fresh Lambda has noticeable latency. So, some people have resorted to periodically querying their Lambda to prevent this. However, occasionally Amazon will reset all Lambdas, which will force the hard restart.

I'm aware of the increased startup latency for functions haven't been recently used, but that's not the same as downtime or dropped connections.

Deploying code doesn't take a lambda function down. It just spins up the next instance with it.

You do have a cold start issue as mentioned above but if that isn't an issue, then you shouldn't see any down time.

We run a beta / prod system to do testing and then for blue / green, we deploy a second function and switch the API Gateway over when we are good to go. Pretty straight forward