BriskBard, CEF4Delphi, WebView4Delphi, WebUI4Delphi and WebUI4CSharp

Try the TChromium.OnRenderCompMsg event and set aHandled to TRUE for the WM_XBUTTONDOWN message.

The TChromium.OnRenderCompMsg event has a known bug and it stops working after the second TChromium.LoadURL call.

If TChromium.OnRenderCompMsg works the first time and your application needs to load several URLs try using TChromium.RenderCompHWND with SetWindowLongPtr to use a custom procedure to intercept the WM_XBUTTONDOWN message.

I don't have one of those mice with navigation buttons but I would try the TChromium.OnPreKeyEvent and TChromium.OnKeyEvent.
Perhaps those buttons trigger a key event that you can intercept in those events.

Sorry, I forgot to add that you have to set aMessage.Result too.

I'm not sure if all the Windows message set aMessage.Result as a boolean and I shouldn't set it in the TChromium's code.

In the case of WM_XBUTTONDOWN try setting :

Try setting a boolean variable to true when TChromium.OnRenderCompMsg receives a WM_XBUTTONDOWN message and then use the TChromium.OnBeforeBrowse event to cancel the navigation if the previous boolean variable was true.
https://www.briskbard.com/forum/viewtopic.php?f=8&t=700

If that fails, you can also use a low level mouse hook with SetWindowsHookEx and WH_MOUSE_LL to intercept any mouse message.