Definitely.
Just $18 for a decent one you can later improve. First Aid Only 299 Piece All-Purpose First Aid Kit (FAO-442) https://www.amazon.com/dp/B000069EYA/
When skiing inbounds on groomers we do not carry anything for us or kids per se. The kids ski/ snowboard with hydration backpacks, and have snacks and a couple of bandaids in it.
For anything else we rely on ski patrol.
We do travel with 2 of these ( 1 always in car, one floating in the bags being used for travel): First Aid Only All-Purpose Essentials Soft-Sided First Aid Kit, Blue, 298 Pieces
So everything below is something I have purchased myself and used. Background: I've worked mostly for cable companies and as a fiber optic lineman so I'm outside a lot and often in isolated areas.
I've started using these for water as opposed to traditional hard canteens. I do still use two quart plastic canteens for most of my water.
Good cheap backup knife, the bright red makes it hard to lose.
Bright small glowsticks, use them to mark people, places, and things.
Backup battery to charge my phone and whatnot.
A large pouch to hold whatever. I keep a couple changes of clothes in one. Food in another, and other stuff in a couple others.
Medium sized pouches.
Large-ish pouch, perfect size for the first aid kit I carry.
Reusable zipties.
A cheap flashlight.
A decent poncho.
Towels, because you're going to get wet and/or dirty.
First aid kit, because boo boos happen.
Knee pads, because sometimes you might have to crawl.
A belt that can hold pouches which can hold other supplies.
Tags that can tell others your blood type in case you can't.
An inverter for your vehicle so you can trade gas for juice.
Something to store matches or other small things you need to keep dry.
A good hammock to sleep in.
Something to cut things with.
A good small multi use screwdriver.
A backpack to carry stuff. This goes everywhere with me and holds enough to keep me going for a day or two.