A ToolStripButton does give visual indication of being clicked, in all display styles.
There is also CheckOnClick which makes it behave like a toggle button to stay 'depressed' when checked, if that is what you mean.
Given that, by default, a ToolStripButton doesn't look like a regular Button to begin with, i.e. it doesn't look 3D, it can't possibly give the impression of being "down" or "up". If you want that then you must change the appearance of the ToolStripButton altogether, which would require that you created your own renderer, which is not a trivial task. The simpler alternative might be to use a ToolBar component instead of a ToolStrip. It has the older look and feel by default that you seem to be looking for.