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.
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