I can't find the broken pickaxe.

[Tominion]

Is there any way to get a broken pickaxe?
I'm asking because I managed to unlock the entrance to the top and I can get to most of the locations without any problems - the problem is that in front of the buried entrance to the mine I can't find a broken pickaxe, which, according to the task, I have to give to Sara to repair it and then able to use the mine, which is why I was "really" stuck with the rest of the quests. The only thing I can do is complete the tasks given to me by the residents, but they are slowly coming to an end.

[Darkness Vortex]

Did you speak to Sara after you investigated the mine entrance? I feel like that one took me a while to figure out too because I did some things out of order...

[Tominion]

Sara keeps telling me to come back later because she's always busy and it's been 3 days (in the game).

[Darkness Vortex]

Hmm... Have you been to the lake yet?

Edit: Also, do you make sure to keep some Inventory slots open in your backpack?