Disable download button for Google Chrome?

Demmongonis solution does work but be aware it can lead to unwanted results.

Android/Chrome sometimes, depends in the video I guess and other factors, adds buttons at the right of the download-button. i.e. the casting-button (there is no way to select it). It will make the download-button to remain visible and the last button to get hidden (casting-button)

Update

It is posible now to hide the download button using the controlsList attribute:

<video controlsList="nodownload" ... />

You can inspect the controls of the native Chrome Video Player by activating the shadow DOM in Settings|Preferences -> Elements -> Show user agent shadow DOM

After that you can inspect the players buttons.

Now the problem is that the download button cannot be accessed via CSS for some reason.

video::-internal-media-controls-download-button {
    display:none;
}

won't work. Even selecting the preceding button and targeting its neighbor using + or ~ won't work.

The only way we found yet was nudging the button out of the viewable area by giving the control panel a greater width and making the enclosure overflow: hidden

video::-webkit-media-controls {
    overflow: hidden !important
}
video::-webkit-media-controls-enclosure {
    width: calc(100% + 32px);
    margin-left: auto;
}

I hope google will fix this issue soon because most content providers won't be happy with this...

or you can simply add nodownload in controlsList

<video width="512" height="380" controls controlsList="nodownload">
    <source data-src="mov_bbb.ogg" type="video/mp4">
</video>