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Let's say the guidance system fails, and Falcon 9 first stage gently touches down on surface of water before switching the engine off (and plummeting into the water).
Will it sink or will it float?
At that moment it's a huge, mostly empty tube, although rather heavy on itself, I'm not sure if it's durable enough to survive a splashdown from 0 altitude, and how much fuel is left after the landing. And while I could find the landing mass, I'm really unsure how to discover the landing volume.
Past events demonstrate that it will sink. Following the ORBCOMM launch in 2014, the first stage performed a soft landing in the Atlantic Ocean, and the water impact caused loss of hull integrity:
After landing, the vehicle tipped sideways as planned to its final water safing state in a nearly horizontal position. The water impact caused loss of hull integrity, but we received all the necessary data to achieve a successful landing on a future flight.
The SRB's, made of much heavier material than the Falcon 9 first stage, would hit the water, tail end first, with the heavy motor casing at the bottom trapping air inside it.
It should be noted, the SRB's did NOT impact gently, they hit at terminal velocity under their parachutes, and usually would bend the SRB segements some amount out of round that had to be fixed in the factory.
This air inside would suffice to keep it floating mostly vertical until the recovery ships could find it. Only about 20 feet or so stuck up out of the water.
To recover it, they would have divers swim down with an inflatable plug, to close the hole the engine makes, and pump air into it until it was floating on it's side, then they could tow it back to Port Canaveral for processing.
You can watch this in a great video SRB Recovery
The Falcon 9 first stage is much lighter and less structurally sound. Additionally, it would come to a 'stop' relative to the surface and then fall down, somehow.
In the soft landing attempts SpaceX made before they had an ASDS they reported that the stage survived landing and was destroyed by wave action once on the surface, which indicates it likely was floating.
It does seem that the fall from standing tall, on the surface, down to the surface likely did much of the damage as well.
The Falcon 9 would survive partially, but would be in pieces. I have heard rumor that the first ocean soft touchdowns of the Falcon 9 that parts of the booster survived, which were later tested when various issues arose with the rocket at a later date. However, the tanks themselves were gone, and I'm sure the engine was rusted to the point of not being easily usable. But it could survive, at least to some extent, as has been shown.
Even if it stays afloat the engines will be wrecked by the salt water. The recovery value would be pretty low.
