Can an Out the Money put option's price $>$ its Strike Price? I'd guess yes, if the put option's IV spikes. An OTM put has merely Time Value and no Intrinsic Value.
As happened to crude oil futures in 2020, prices of certain traded instruments can go negative. In such a case, the strike can be at zero, and the put has positive value, even if the spot price is still positive.
Otherwise, it is only possible if:

interest rates are negative, and so one is willing to pay money now for less money in the future, or
somebody hit the wrong keys when trading, and it was not overridden by automatic controls.

In a positive interest rate environment, fair value is always less than the strike price if the price is floored at zero, since the maximal payout is the strike. Increasing volatility cannot affect that.