An estimate can be made; others have suggestions. Here's a simple one.
If you know the height from which it is dropped, then you can calculate its speed when it hits the ground. The ball will be deformed, reaching maximum deformation when it's speed hits zero and the ball starts to go up.
For a crude estimate assume that the ball deforms by, say, 1/3 of its diameter. That is, the ball continues to move downward for and addtional length of 1/3 of its diameter after the ball first makes contact with the ground. That fraction could be 1/10, or 1/2, or 1/10000 for a metal ball, but we have to assume something. Assume also that the ball's acceleration is constant once in contact with the floor. That distance, along with the speed of the ball, gives you the duration between first contact and maximum compression. It has to go back up, so multiply by two.
Without a lot more information about the ball, and potentially very hard math, that's about all you can do. Any assumptions you make about the ball's elasticity are guesses, so it's not worth the effort. I suspect that the estimate you get this way is not a bad estimate at all.