I've seen this many times as an options market-maker: A short-seller fund can't get borrow (which is a requirement for short selling) on a name it's already short, so it can't add to the short position. Thus, they call a bank and buy put options. The bank makes a price that incorporates (charges for) the borrow risk, usually because (A) the bank has an easier time locating borrow and gets a better price, (B) the bank trader/desk has a larger tolerance for P&L swings and can thus take the other side of the trade and hedge the position slowly or not at all, or (C) the bank has better market access and can trade combos (synthetic shares) against a different institution that has stock or a better borrow rate.
As a general rule, there's options activity for names that are very volatile. This doesn't always show up on the open interest count, because many firms trade OTC contracts that don't have the same clearinghouse protections as exchange-listed option contracts.
Option bid/ask spread being wide don't necessarily mean that volatility itself is illiquid. There could be large size on both bid and offer, with every dealer under the sun making a market, but the carry rate spread is wide, and that widens the option price. For example, the put offer will be based on the short rate, and the put bid will be based on the long rate, so if these two rates are divergent (as is the case when a name is hard to borrow), then the spread will be wide even if both sides are priced with the same implied volatility.
BTW even if screens show little more than a 100-lot, a bank will make a price on a much larger block of options. You won't know where liquidity is without quoting dealers. An option market could show a bid/ask that's 2 dollars wide with 200 contracts on either side, but a market maker may make you a two-way price that's 30 cents wide for 10,000 contracts. Depends on a lot of factors (who's the client, what's the position, does the market maker know the name and have a view, what are the catalysts, why is the screen market so wide, etc)