A Tecnam P2006 Flight Instructor reported a tail strike on landing; due to an airspeed anomaly; caused by a malfunctioning Pitot-Static system.

Narrative

I was acting as pilot in command (PIC) and Flight Instructor of a Tecnam P2006T during a training flight for an MEI add-on course. We were doing pattern work at ZZZ where we did 2 landings. On the first takeoff roll on Runway XXL (full length); the Primary Flight Display (PFD) indicated our rotation speed of 65KIAS abeam Taxiway 1 which is roughly 2;000ft down the runway. Looking back this is an abnormally long ground roll as our calculated ground roll was less than 1000ft. On climb out; after gear up; holding Vy (84KIAS) our rate of climb was ~400fpm (flaps T/O) which raised an eyebrow for me because usually our rate of climb in this configuration is ~600fpm and it was cooler outside so performance should be better than usual. However; once we went flaps up at 400ft; rate of climb went up to ~800fpm which is around normal so I didn't think much of it. Looking back on ADS-B trackers our groundspeed on climb out was over 110kts with calm winds. The first landing was a normal landing and everything felt normal; the student landed well. On the second ground roll; it did feel a little longer than usual again; but not so much to cause concern; so I did not think much of it. On the second landing; we did a short field landing; which requires coming in at a slower airspeed. When on final for the short field landing; we were ~75KIAS; with flaps to the landing configuration; however the stall warning horn was chirping intermittently. This being said; I have squawked this aircraft in the past multiple times for early stall warning horn activations at unusually high airspeeds; so when I saw 75KIAS on the PFD and the stall warning horn chirping; the first thought that came to mind was this was another instance of erroneous stall warning indications. Anyhow; we continued the approach; and on short final; I noticed we were at about a ~5 degree pitch attitude; which is abnormally high for flaps landing and 75KIAS. As we crossed the threshold and the student brought power idle; the plane immediately dropped; the student out of reaction tried to flair to save it; but because we were already at a high pitch attitude; the flair caused us to tail strike. I took controls; but because we dropped so quickly and; less than 5ft off the ground; I was unable to mitigate the tail strike. On landing roll out; the IAS showed 30KIAS and stopped there. Even when we came to a full stop; the PFD still showed 30KIAS. I cycled the alternate static switch and the airspeed dropped to 0. Looking back; there were several safety factors; that resulted in this incident. First off; with the incorrect airspeed indications. After post incident inspection by an A&P mechanic; it was found that the static lines were heavily water logged. This could have been prevented by routine suctioning of the static port; especially since we are in such a humid and wet climate in the summer. In terms of the stall warning horn; I should have not assumed that it was an erroneous indication and took it seriously. Going forward; I will take the stall warning horn seriously to mitigate safety risks regardless of if I believe it's an instance of an erroneous stall warning indications.