Is using Lambda with an external data store (like MongoDB) a solved problem?

I remember this being an issue a long time ago, where Lambda pretty much was a non starter for use cases where you're running 1000 independent tasks that each, on their own, is a small unit of work (just 1 or 2 database queries).

Long way of asking: is there a way to share a database connection between lambda functions or does each run need to re-establish / re-authenticate a new connection with the database?