That depends on the provider and the database. For AWS Lambda with DynamoDB connection pooling is handled "behind the scenes", and you don't have to worry about it. AWS supports SQL databases as well, I would assume it also is pooled for you.

NOW currently doesn't have databases and tells you to use cloud DBs. So more than likely the connection is dropped when the lambda shuts down.

But if you'll let me go a little bit meta, the idea of serverless is that you shouldn't even be thinking about things like connection pooling. If you just use the databases that your provide supports those types of concerns are their concerns now, not yours.

In other words, if you want to pick your own tech stack then serverless is not for you (yet). If you're willing to surrender your opinions and just use what the platform provides you (if you're using AWS just use DynamoDB) then you can stop thinking about these problems and focus on your business logic.