We have had problems getting the anchor lamp to work for a couple of seasons. So with the mast down we were able to inspect the fitting in detail.

It turned out that the plastic base was cracked (probably letting moisture in) and the plastic clips that secure the bulb holder were broken (almost certainly preventing any reliable connection to the bulb.

A lucky find on eBay – the exact part: a HELLA all-round white 10W (old stock) masthead light. Condition as new, and supplied complete with a LED bulb.
Many thanks to Marine Miscellany for the supply.