Since there are many answers saying "No", I'll provide anecdotal evidence of a "Yes" answer.
I was flying from Indianapolis to Klamath Falls, OR via MSP and PDX. As I sat in MSP, they announced a delay of the flight to PDX. I politely inquired at the gate to see how long it would be and to check on the likelihood of my making my connection at PDX - they assured me there would be no issue. As I sat there, they announced delay after delay. Each time, I went back to the gate agent, slightly more concerned than the last time, asking about making my connection at PDX, since I knew I was already on the
last only flight of the day from PDX to Klamath Falls. They continued to assure me that all would be fine.
On my last trip to the gate agent after the last delay announcement, I was met with, "have a seat, give me a few minutes and I'll get back to you". After being called back up to the gate I was assured that I'd make my connection.
Upon arrival at PDX, I stood up in my seat at the very back of the plane and looked forlornly at the many, many rows of not-moving people in front of me wondering if I'd ever get off this plane, to say nothing of making my next flight. I heard mechanical noise behind me and saw that the rear door had been opened. I heard a flight attendant say "I think it's the guy in the white sweatshirt". Sure enough (after checking), I was wearing the only white sweatshirt around.
A guy in a ground-crew uniform tapped me on the shoulder and asked me to follow him. He lead me out the back door, down the stairs, across the tarmac and to the front of the security line for the terminal where my flight was departing. He said, "Hurry, you're the last passenger and they're holding the flight for you. When you get to your destination, contact the luggage people and they'll get an address to forward your checked luggage."
As I boarded the otherwise packed puddle-jumper, the FA welcomed me aboard and closed the door behind me.
So yes, they may well hold the plane for you, and even get you through security (if necessary) in a hurry.
*NOTE: all conversations are approximate - this was 25 years ago on Northwest Airlines (which no longer exists). YMMV.