Can I copy part of a Minecraft multi-player (SMP) map to a single-player map?

You could ask the server admins. Many times they will zip the map and send it to you if you ask nicely and explain the purpose.

You can use World Downloader.

Press Esc in-game and press L to start downloading. It will download everything you see — about 160 blocks away from you, from bedrock to the sky limit. To stop the download, press L again.

Then, just look in your single-player maps.

It's theoretically possible. Your client will download part of the map before rendering it, so the data is actually transmitted through network and stored in memory, and is accessible by either packet capturing or client modding. Either way is straightforward to implement, and I implemented the latter one with MCP before for a certain complex labyrinth map.

However, such tools for retrieving server map can be easily used for hacking and unfair gameplay. So don't expect any legitimately published ones.

Edit: I came across an implementation called WorldDownloader which may be good for your need. You can try it out.