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.