If you buy a put, and the stock goes to 0, you gain the strike price, and stock prices cannot go negative, so it makes no sense to buy a put for more than the strike price.

The person/bot with $5.20 as the ask is merely being optimistic, which costs them nothing. Maybe there will be someone dumb enough to pay that, and they get free money. It's like saying you have apples for sale for $1 million each. You won't sell many at that price...but you only have to sell one!

At a guess, the gap is unusually wide also because it's a relatively obscure security or time frame, so the contract trades rarely if at all and nobody else sees it worth the bother of offering them for sale at a better price to undercut the first optimist. Such a not-quite-so-greedy person might show up in the split second that a bid appears.

You are buying the right to sell something for $5 in future. Its current market price is $2.24. Clearly that right is worth $2.26 today, even before you consider the time value. So if I wanted to buy it, I'd put in a limit order at a price just a hair above $2.24, and inch it upwards till it fills. That's pretty much how I trade all my option contracts, and I've done tens of thousands of them. It would likely fill as soon as the rate of return on the time value is considered to be within the reasonable income expectation range of the market maker. It will almost certainly fill long before your bid gets up to $5.20. (sooner if it's a reasonably large or liquid stock, maybe painfully higher if it trades by appointment)

Jim