A C172RG pilot reported loss of control and runway excursion when his 'special needs' passenger apparently got on the rudder pedals; causing loss of directional control.

Narrative

I took a 'special needs' [passenger with a] 4th grade IQ along. Before departure; I did the routine pre-flight and heavy explanation of what to expect and what NOT to touch before departing. The direct flight is only 39 NM away and had a wonderful smooth ride along with [passenger] enjoying the experience! Upon final and near 'ground effect' over the numbers; [passenger] began getting nervous and grabbing my arms. Focusing on 'flying the plane' it was not an issue for me...then on roll out and putting up the flaps; I began to have no control of my pedals. We began to go slightly left and still had no control. I fought and fought to get control while slowing her down but unable to. Eventually we went off the left side of runway into somewhat of a ditch into mud due to melted snow. [We did] a nose plant into mud; causing prop strike; blown left main; right wing tip scrape and dent; and nose strut and fork eventually snapped. After this all 'settled' I was wondering how and why this happened; so that I can avoid this from ever happening again? That evening I was going through the possibilities...did the pedal cable snap on me; did the left main blow upon landing; or did [my passenger] shove his feet into the pedals and not know it being so nervous even though he told me three times he had kept his feet near the seat? The more I thought about it; the more I leaned towards [my passenger] not knowing due to his nerves along with being a special needs young man shoving his [feet] into the pedals on his side. [Insurance adjuster and I] went out to the runway to investigate. After about 15 minutes of observation of left main and discussing the scenario; this is not a scientific conclusion to be sure; but we agreed that this is 95% likely what took place. To avoid this type of 'incident' ever happening again; I'm looking into removing the passenger pedals; or [possibly] stow them when flying new passengers.