I just installed a Peerless Kitchen Faucet Model Series 99596 and the sprayer drips water when the water is turned on and water is coming out of the spout.

All my connections are tight and no water is leaking anywhere else. Do you think this is a faulty sprayer?

Or is it a problem at the diverter?

The problem is in the hand held sprayer... take it back to where you bought it and have them give you a new one. The only thing you have to take back is the spray head that unscrews from the hose, not the hose or the other parts that stay on the hose.

Letmetellu is bang on.
Once you presserize the faucet the sprayer comes under pressure also. All a diverter does is shut off pressure to the spout when you open the sprayer. Follow his advice and exchange the sprayer. Good luck, Tom