As a matter of further investigative work, my next step would be to confirm that what appears to be a single paragraph in the table cell, really is one. Do this by displaying hidden characters. On the Home tab of the ribbon, select the backwards P symbol in the paragraph group. This usually hidden symbol ends every paragraph. If for example, the first line in your table cell ends with it, that's a paragraph, so you will not see any hanging indent applied.

Assuming that's not the issue, I would try applying a style with a hanging indent to an existing paragraph in your table. If you have never created a style and are unsure about it, let's use Format Painter to see if that will apply the hanging indent.

Using a paragraph outside the table, format it with a hanging indent. Then, with your cursor still in the paragraph, either click on the Format Painter icon in the Home tab (the paintbrush, at left) or press Ctrl+Shift+C to copy the formats of the paragraph. Then, if you used the icon, click in the table paragraph, or, if used the keyboard shortcut, place your cursor in the table paragraph and press Ctrl+Shift+V.

Any luck?