Spiegel joins on a one-month loan deal.
"I've watched him play a couple of times and we've been given a lot of footage to review as well," Carlisle goalkeeping coach Tony Caig said.
"He's a good size, he's mobile and vocal, and he's a Swiss Under-21 international so he has good pedigree."
He added: "West Ham are looking for him to make that next step and an opportunity to get out and play some football provides that opening for him."
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