Thank you for your question.
Please could you tell me whether you agreed verbally on the conditions that apply to the deposit?
Did you issue a receipt? If so, what did the receipt say?
Did you have email contact with them where you mention the deposit? If so what does this email correspondence say?
I look forward to hearing from you.
Also does it say anything in your email exchange about the time limit for cancelling the reservation?
From what you tell me, it is clear that the clients agreed to make a booking in accordance with your terms and conditions. Therefore the fact that they did not sign the contract is of no consequence. The contract exists between you on the basis of your email exchange where you made your terms clear and they accepted those terms.
If their cancellation does not respect the deadline specified in your terms, then you are entitled to retain the deposit in accordance with your terms.
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