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The BBC page and Sony's own support page both confirm that from 17th February, 2020, models such as my KDL-W605A are no longer certified to run iPlayer, and therefore the service will no longer be available. Given that the built-in iPlayer capability was the main reason we bought this product, I am rather annoyed that Sony won't upgrade the firmware to continue supporting the iPlayer.
The "devicehelp" page suggested by the Beeb was not particularly informative. However, I did some research and discovered a £28 device available from Curry's/PC World that should compensate for the loss of this facility: the ROKU Express 2019 HD Streaming Media Player. I have ordered one and once I have tried it out, I plan to report here how it worked out.
I'm pleased to say that the Roku Express set-top box, which is about the size of a two matchboxes, worked perfectly. It gets enough power from the USB port at the back of the TV, comes with both a USB and a HDMI cable, and was instantly available from Currys for a mere £28. The only fiddly part was to log into our home WiFi network, because it seems it could not use the WPS system - I had to type in the long WLAN key to get connected.