It is easy to construct an unconditionally secure encryption scheme where the key is reused to encrypt multiple messages. Simply take any unconditionally secure encryption scheme that can be reused $1$ time and then to generate keys sample $m$ keys of this original scheme (e.g., $m$ OTP keys). Now you can use these $m$ keys to encrypt $m$ messages and obviously the security is still that of the original scheme.
Of course, you may argue that this is 'cheating', because now we have an encryption scheme with keys of size $n\times m$, where $n$ is the size of a message. However, this cannot be helped. For the scheme to be unconditionally secure the key must be at least as long as the total amount of bits we can communicate using it. To see why this is the case consider that to have perfect secrecy given a series of ciphertexts any series of messages must be equally likely. This means that for each series of message bits (combining all the messages sent) we need one distinct key, i.e., the key must be at least the size of the total amount of bits encrypted.