Since a few more posts have arrived on this topic I thought it would be
a good opportunity to summarise progress. My thanks to everyone who has
sent me information.

As a reminder the photo is here:

https://britastro.org/cometobs/1965s1/1965s1_19651102_0230_unk.html

The comet's position and the horizon orientation indicate that the photo
was taken in southern Africa at around 02:30 UT on 1965 November 2. It
was then a morning object. The large field of view implies that this was
taken with a short focal length lens on a 35mm or 120 format camera. The
smeared horizon detail indicates that the camera was driven at
(approximately) the sidereal rate although, as noted, the stars and
comet head are trailed a bit. The length of the smeared trees implies a
driven exposure of 6-8 minutes as the comet rose.

Auke Slotegraaf sent me a list of the plates taken at the Royal
Observatory at the Cape but none of these really fit, either the timings
or the field of view. The plates were taken for astrometry so were much
larger image scale. Tim Cooper checked through the archived Monthly
Notes of the Astronomical Society of Southern Africa for any reports of
observations or images of the comet reported to the Comet and Meteor
Section, at that time under the direction of S C Venter.  The only
visual reports recorded in that journal were by Jack Bennett and S C
Venter.  Neither mentions any photographs.

Tim suggests that the photo could have been taken from what is now
Zimbabwe (then Rhodesia). Martin Mobberley pointed out a couple of
reports in the JBAA (76/2 and 76/3). In particular the second one says:
"After thanking Mr McKie, the President called on Mr. Lyons to show
three colour photographs of Comet Ikeya-Seki taken from Rhodesia with
Kodak Retina IB using Kodachrome II, the exposure being 10 minutes at
f/3.5. Unfortunately the photos are not reproduced in the Journal. Does
anyone know who Mr. Lyons was?

I hope that one day I'll come across the answer to this mystery
somewhere in the dozens of boxes of the section archive.

Nick.