How can I include only part of an image caption in a table of figures in Microsoft Word?

It turns out the solution to this problem in Word 2003 and later is to use something called “style separators”. Style separators are hidden characters that are functionally paragraph separators, but they don't cause a line break; like this:

The quick brown fox│̲̅¶̲̅│jumps over the lazy dog.¶

They allow a paragraph — or rather, what appears to be a single paragraph — to have multiple paragraph styles.  This is analogous to the way a “section break (continuous)” is a section separator that doesn't cause a page break, thus allowing a page to have multiple section formats.

Firstly, make sure you have enabled viewing hidden characters.

To insert a style separator in Word 2007 use the keystrokes Ctrl+Alt+Enter, or add the “Style separator” command to the Quick Access Toolbar from the “All Commands” set (or the “Commands Not in the Ribbon” set). When you insert a style separator, it will be added at the end of the paragraph in which the cursor is placed. Move it to the point at which you would like your caption to be split.

Or, in other words ^{(or not?)}, at least in Word 2013 and 2016, “insert style separator” is a misnomer.  This function (shortcut/button) simply changes the following paragraph marker (at the end of the current paragraph) from a standard paragraph mark into a style separator, thus joining two paragraphs into what looks like a single paragraph.  The best way to work with this might be to type Enter where you want to break the paragraph, then click to the left of the new paragraph mark and use the shortcut.

Don't forget to select all (Ctrl+A) and then update field codes F9 to make sure your changes get propagated into the table of figures.