Lancair Evolution pilot reported pitot static system icing in climb. The pilot diverted and landed without incident.

Narrative

Flew my plane from ZZZ to ZZZ1 to have lunch and get the plane washed.After lunch and wash; preflighted plane; filed to fly to ZZZ2 at FL270 (Note: It was in the mid 90sF at ZZZ and the low 80s in ZZZ1).Departed ZZZ1 and activated autopilot.Conditions were clear and at FL180 I turned the pitot heat on per check list.I did note that when I turned the pitot heat on; OAT temps were already -10c and thought nothing further.As we approached FL240 the plane began to pitch up nose high and airspeed dropped to zero.I disconnected autopilot; hand flew the plane and began to trouble shoot the issue. Since the pitot heat was on from FL180 to FL240 (2 mins?) I incorrectly assumed that the heat caused an issue. ATC cleared me for FL270 and my copilot correctly told them unable.We checked in with ATC and told them we lost airspeed and altitude indications. ATC said MODE C was reporting an altitude that we knew was what the airplane was showing but no longer correct. We knew we had a Pitot static issue but needed to determine why.My pilot/passenger correctly suggested that the appropriate action was to actually leave the pitot heat on and he was ultimately correct. We got a little airspeed indication on the instruments but was not enough to feel comfortable so co-pilot requested priority handling with ATC. We received vectors to an arrival procedure back to ZZZ. After about 5 mins all instruments returned to normal and we cancelled the priority handling. I landed without incident. ZZZ Airport Fire and Operations met us at my parking space to make sure we were OK. We suspect that the cause of the incident was water in the pitot tube from getting the plane washed at ZZZ1 exacerbated by me not turning on the pitot heat before the water in the pitot tube froze.Consider always flying with pitot heat on. Make OAT a more regular part of my scan