Extra deposits for Pet Damage?

Tenancy Deposits

We understand that deposits are capped at 5 weeks rent and we're OK with that as we stipulated that we didn't want tenants with Pets so hopefully any damage would probably be covered in a deposit that size.
We've started adverting for a new tenant and had a few applicants who look really good but they do have pets and have offered a higher deposit. We've told them we can't legally take a higher deposit, but is there anything that we can do to protect ourselves from being out of pocket if we decide to let to someone with pets?

Can we write a clause in that they have to cover any additional damage in excess of the deposit?

cheers