micro switch sensitive to impact
Lots of things inadvertently display some effect from a mechanical impact- when we don't want it to happen we call it microphonics. 

Things that are intended to respond to mechanical effects are called microphones, accelerometers or shock detectors (and probably other things I can't think of at the moment). 

If your intention is to detect an impact on a target, then you probably only need sensitivity in a single axis. Here is a piezo shock detector, and below is an example circuit (the actual detector here is designed for alarm systems and is probably too sensitive for your application). I would certainly add a series resistor to the 4093 input because there might be a fair bit of energy from a piezo transducer. You could add a monostable multivibrator (one-shot) to the output to provide a short "switch closure"). 

A piezo speaker or magnetic transducer (beeper without circuit) would probably work fine too. They're not designed for the purpose, so you'd want to hook them up to an oscilloscope and see what kind of output you actually see from the device when used as a sensor. 

There are also electromechanical sensors used in applications such as alarm systems and sneaker (trainer) shoe lights for kids that consist of a normally closed switch with some kind of a weight that makes the switch open when it impacted. It's possible a normal reed switch capsule would work in this application, but it won't be 4 x 4 mm (more like 50mm x 2mm diameter). 



Here is a smaller SMT one (here) from Kyocera, but still a bit bigger than your stated requirement (4.8 x 1.7 x 0.8mm).