Hello Harel!
I have a similar desire, small computer as presenter. Not for projection, but for diplaying on big TVs. I tested two solutions so far:
1) Asus VivoStick, running Windows 10. Requires an external keyboard/mouse/touchpad solution, a power supply (included), it does not run on a battery! Unit has no internal display, you must use the projector/TV as monitor, awkward in some public situations.
Very small though. Connects with regular size HDMI. Has worked well for me up to full HD resolution. Doubtful with 4K. You must be careful to not further clutter up the internal storage, suppress updates, throw out garbage that came installed etc. Performs far better for me than some customer reviews suggest, but probably better suited to be attached to the rearside of a TV permanently, comes with a holding device for that purpose.
2) A windows tablet. I have an 8"-screen Acer Iconia W4. Great battery time, quite a few hours, even after some years of usage. The on-display keyboard should be all you need for work with presentations. Has HDMI (micro) outlet. Here's the drawback: Got it nearly four years ago and for another purpose. Resolution is below full HD!! Looks quite ok on TV when you run a film, but today one would want at least 1920 x 1080 output.
Check out Acer, Asus and other brands for Windows tablets with fullHD output (other Operative platforms are useless for me).
The NuTech (oops!Chinese!) brand has interesting 8" tablets with full HD. Generally fairly good customer reviews, except for WiFi performance, lagging! Attractive price though (80-90 $ at some vendors).
https://www.windowscentral.com/nuvision-review. This tablet I haven't tested, but consider it myself.
Good luck!
Wolf