1987 BMW 535i BackFiring

Hi All,
I have a 1987 BMW 535i and it's been running very well up until the last few months:
The engine started backfiring and then stalling momentarily a few months
ago. This only happened once in a while so I kind of thought it might be a bit of water in the gas tank. The same problem gradually got worse over a period of time and now it's happening all the time. A pattern has emerged as follows:
The engine runs fine for a while and then out of the blue it starts to backfiire and then stall. I then pull off the road and sit for a few minutes. I then proceed to start the vehicle back up and everything works fine for another few kilometers or so. Then the backfiring/stalling happens again and the pattern repeats...again and again..
A mechanic I'm not, however, could it be the fuel pump as it's never been changed in 320,000km.
Any feedback would be greatly appreciated.
Thanks in advance.
Add pictures here
<% if( /^image/.test(type) ){ %>
<% } %>
Add image file

Motorsforum.com is a website by car enthusiasts for car enthusiasts. It is not affiliated with any of the car or spare part manufacturers or car dealers discussed here. All logos and trade names are the property of their respective owners.