The DM should not have asked the players for a roll if they didn't like the outcome of a failure. If the DM wanted to narrate some mechanical effects from this perilous journey, a better choice might have been "roll to see if you suffer a level of exhaustion" or "roll to see how much cold damage you take from the blizzard". However, assuming that moment has passed, here are some things to try:
- "There's no problem, because like any sensible climber I have secured myself to a grappling hook before climbing, and it catches me before I fall too far."
- "I pull out my grappling hook and throw it at the cliff to catch something."
- "I use my fast reflexes to catch a rock before I fall off the cliff."
- "Well, I roll past Susan on the way down. Susan, have you got a way to catch me?"
- "Okay, after Susan I roll past Frank. Frank, have you got a way to catch me?"
- "Well, I know that Frank has a feather fall spell. Frank, wanna cast feather fall on me so I don't die?"
- "Well, rolling down the slope isn't the same as a straight fall. I brace my shield against the slope and sled down it."
- "I turn so I'm falling shield-first and I use my constitution to take the impact on my shield."
Many of these answers are sort of bullshit -- in particular the "use my constitution to take the impact on my shield" thing is pretty far outside the scope of what the rules should allow -- but, if the DM is looking for an excuse to not kill the dwarf, this sort of thing is more plausible than "I flap my arms really really hard".
It's outside the scope of DM responsibility to make up tactics for the character trying to survive. If the DM starts narrating the character doing stuff, the whole table will feel that it's fake.
What the DM can do is tell the group how bad the consequence of failure is. "Okay, you fall sixty feet onto a rocky outcropping. Take 6d6 damage. Does that kill you? No? Okay, what does everyone do now?" Sure, it might have been implied that falling would be fatal, but that's not actually been narrated yet and the DM can still change it.
It may interest you to know that I've had a character in a very similar situation. We were climbing down a rope, and the DM narrated a xorn stepped out of the cliff face and cut the rope above us, and he asked: "so, for a split second you're hanging in midair, before you fall. What do you do?"
We couldn't think of anything. We fell the distance and took the falling damage, and then did the fight. After the fight was over, I realized I could have given the "pull out my grappling hook" or "catch onto a rock" answer.
Later, I had a different character in this situation. We were on a platform, and the DM told us that the platform was unsteady and we needed to make checks to not fall off it. My character failed the check, so the DM told me I had to make a different check to catch the edge of the platform. My character failed that check too, so the DM told me to make a third check to catch some other thing. My character passed that last check. All the while I was thinking: "well, are you going to narrate that my character died because of failing a check, or aren't you?"
After that, I started having all my magic-user characters carry a scroll of feather fall around, just in case. Rules As Written, you probably wouldn't have time to fetch a scroll of feather fall from your backpack while falling -- but, if your character is about to die, the DM might be willing to ignore the Rules As Written.