First battery was DOA, bought a second one and it wasn't DOA, but does seem to have an issue:

Press round button - round light on solid one light bar flashing - goes off.
Press and then press again and hold - round light on, light bars come on in sequence and then just one light bar on for a moment - goes off.
Plug in to charger - round light on, light bars come on in sequence and then one lgith bar flashes quickly for about 11 or 12 flashes and then it flashes at a slower speed for 2 or 3 flashes - goes off.

Which LED is flashing when charging? Please try to keep the battery in the charger for about 2 hours.
For power on the battery, please press and press and hold for 2 more seconds, don't let go of your fingers too fast.
If the issue persists, please provide a video of this case for a better help. Thank you.

It sounds like your Battery may be fully discharged, so following DJI Thor's advice should bring it back to life.  However, it is important to know what lights are doing what as they do have indicator codes that can explain what the problem is.  See the diagram below to determine what your battery is doing or not (ref P4P manual v1.4 pg 34):

Video in next post. Have checked LED numbering in the manual to add to the info from my first post, see below:

Press round button - round light on solid LED 1flashing slowly - goes off.
Press and then press again and hold - round light on, LEDs come on in sequence 1-4 and then LED 1 flashes once - goes off.
Plug in to charger - round light on, LEDs come on in sequence 1-4 and then LED 1 flashes quickly for about 11 or 12 flashes and then it flashes at a slower speed for 2 or 3 flashes - goes off.

Video -

coupi Posted at 2018-5-17 13:05
Video -

Have you tried leaving it on the charger for at least 2 hours yet?

You should reply to Thor directly so he knows you've uploaded your video.  
You either have a discharged battery or it has been discharged for too long and has gone bad.  However, until you've attempted to charge it for a couple of hours that is not a certain determination.

DJI Thor Posted at 2018-5-16 21:10
Which LED is flashing when charging? Please try to keep the battery in the charger for about 2 hours.
For power on the battery, please press and press and hold for 2 more seconds, don't let go of your fingers too fast.
If the issue persists, please provide a video of this case for a better help. Thank you.

Have uploaded video and specifics of which LEDs flash and when

coupi Posted at 2018-5-19 05:01
Have uploaded video and specifics of which LEDs flash and when

Sorry for the delay. It was the LED 1 flashing, if the issue persists even after charging it for 2 hours, you might still need to send the battery back.
Sorry for your difficulties. Please contact our support for a further help: http://www.dji.com/support

DJI Thor Posted at 2018-5-22 23:35
Sorry for the delay. It was the LED 1 flashing, if the issue persists even after charging it for 2 hours, you might still need to send the battery back.
Sorry for your difficulties. Please contact our support for a further help: http://www.dji.com/support

No problem for delay!

Problem I have is cost of sending back to DJI and then the chance of you saying it will cost me - for a new item? If I were to provide serial number can you not see age/use and determine FOC repair then?