If you're struggling with input lag, even noticeable when not playing online these try these fixes.
Online lag vs. input lag
Most commonly, lag occurs when playing a game online. This is due almost exclusively to your internet connection. Input lag can happen whether you're online or not and is specifically the latency between a button press and what happens on screen and can make games, especially those that require quick reflexes or precise timing such as shooters or sports games nigh unplayable.
Easy Fixes for PS4 Input Lag
Change TV to Game Mode
Changing the TV’s picture setting to Game Mode, a setting specifically made to combat input delay should be the first thing that’s done. Every modern TV has this setting. In some cases there is even a PC mode which may cut down the lag even further.
Charge the controller
The lower the charge the more risk for a bad connection. A better charge equals better connectivity.
Change the connection to USB
Select (Settings) > [Devices] > [Controllers]. Change Communication Method to USB from Bluetooth.
Reset the DualShock 4
There is a tiny button on the back the PS4’s DualShock controller that can reset it. By using a straightened paperclip the reset button can be reached. Turn off the PS4, reset the controller and then reconnect using a USB cable and turning the PS4 back on. Ensure that the communication method of the controller is set to USB (it is defaulted to USB.)
Move interfering Bluetooth signals
Signals from cell phones, speakers and subwoofers and most notably laptops tend to screw with the signal of the controller to the console. moving these from a nearby position, or at the very least anywhere near the signal path can reduce input lag.
Restart the PS4
Sometimes a simple restart can do the trick.
Advanced or Expensive Techniques
Buy a new controller/HDMI cord (or at least switch some around.
Things get old, especially electronics. Trying a new one (or an alternative) can fix the problem, but it could be expensive.
WiFi Router Channel
This is perhaps the most hidden reason that can impact your controller input. Each WiFi router is running on a specific channel. Surprisingly, this channel can interfere with the PS4 input. In most cases this is set to Auto and will need to be changed to a specific channel for the interference to stop. You can call your cable company to help fix the problem but it may be easier to access within your Internet Service Provider’s app. In the advanced settings the WiFi channel can be changed (I’ve had luck with channel 11). The difference in lag was night and day.