here is a little How To Fix BattleEye Client Not Responding Tutorial for you.
Step 1: Close Arma 3 and shut down steam.
Step 2: Open fallowing path: "C:\Users\"USERNAME"\AppData\Local\Arma 3\BattlEye".
Delete the files "BEClient.dll" and "BEClient_x64.dll".
Step 3: Open fallowing path: "C:\Users\"USERNAME"\AppData\Local\BattlEye"
Delete the file "arma3"
Step 4: Open Steam and start Arma 3 again. Connect to your favorite server by Eiernacken and enjoy the bright side of life!
For any questions about topic, feel free to contact me in TeamSpeak.