CIA or Paula?
Unlikely to be a CIA fault as this only reads the left mouse button/fire input, nothing to do with movement input despite some common misunderstanding. You've already said the mouse buttons work OK.
Unlikely to be a fault of Paula as this device only reads in the four analogue paddle inputs (including right mouse button). Again, you've said the buttons work.
Checking that the +5V supply is active is a good suggestion. This is on pin 7 of both 9-way connectors. You can use a bare metal part of the chassis or pin 8 of the mouse/joystick connector as a ground/0V reference.
In the A4000T this 5V supply is via polyswitch fuse F160 on the main board. It's located on the edge of the board, between Paula and the keyboard connector. These devices are essentially PTC thermistors in that they're near short circuit under normal conditions. When fault current flows, they get hot and exhibit a high resistance, thus limiting current. They revert to a normal low resistance state when the fault current is cleared and the device cools, hence they're often called self-resetable fuses.
If your main board is an A4000D (in a tower case), the 5V supply for the mouse/joystick ports is via polyswitch fuse F175, which is the large yellow disc shaped device behind the serial/Centronics ports.
If it's an A4000D that has had major RTC battery corrosion damage, it's often shift register U975 at fault that causes no mouse movement.
I wrote
this article on mouse port repair which may help further.
