In 2005 it also happened to me as I lifted off in my Tri-Pacer: the indicated airspeed slowly started showing lower and lower despite all the rest of parameters (RPM, climb rate, and all sounds were totally normal), so luckily I recongnized the indicator was failing. It eventually marked zero when I was reaching 3000 AGL. Rather than returning to land immediately, I elected to keep flying by the seat of my pants, paying attention to the available instruments and sounds, and a valuable tool I had on board but had not considered even turning on for that particular flight: a handheld GPS, where I locked to ground speed shown. Instead of returning and land immediately, because it was a Sunday and all shops were closed, I changed my destination to a nearby turf field where I knew a little airshow was taking place that day, in the expectation that mechanichs would be around. I approached at about 80 MPH GPS ground speed trying to build a safety net above the normal 70 MPH IAS approach speedand landed there just fine with a roll not longer or shorter than usual. Unfortunately I found no mechanics, so just enjoyed watching the air show and later in the afternoon returned to my base airport, taking off, flying and landing again uneventfullly without indicated airspeed, just paying a lot of attention to the aforementioned variables. Thanks God the weather was perfect and the winds seemed just a breeze and the old Garmin 195 GPS I then had but seldom turned on, had batteries and "found" enough satellites.
On Monday the instruments technician found the butt of a bee that had penetrated the pitot tube and had been pushed through the piping all the way until getting close to the instrument itself!
Although I do not remember being particularly frightened, what would I do different if this happens again? Once landed, regardless of how rural and inconvenient that field is, not take off again until the instrument is fixed.

There's a lot of instrument redundancy built into most aircraft.

Several years when taking off from our base (international) airport, the airspeed indicator dropped to zero -- the pitot tube flap stuck in the down position. Having accumulated about 300 hours of PIC experience prior to this incident, As I performed a quick scan of the other instruments, looked out the windscreen, and tested the control feedback, I confirmed I was taking off with full aircraft control. The end of the runway was too close to attempt a stop.

Again I emphasize the importance of observing the control feedback and the behavior of the aircraft -- because everything else looked and felt normal despite the absence of the air speed indicator.

I contacted the tower (a half mile away from the runway), declared an emergancy, explained the problem, and requested a circle to land. I declined the emergancy equipment rollout. The tower closed all three runways and cleared the taxiways to prepare for my return.

Simultaneously listening to the engine, avoiding the stall warning horn, looking outside, keeping the tower informed, and checking the functioning instruments -- keeping RPM at or above 2400, I climbed to 500 feet (with tower's approval), flew a low pattern, and landed no flaps without incident in the first third of the runway length.

Tower had given me approval for a full stop on runway to remove the pitot tube flap, return to the cockpit, and takeoff.

Lesson learned: Keep calm. Use all available resources. Don't rely on an automatic pitot tube flap to keep insect out of the pitot tube; use a removable cover instead. Keep calm.