We define a p-adic character to be a continuous homomorphism from 1 + tFq[[t]] to Zp. For p > 2, we use the ring of big Witt vectors over Fq to exhibit a bijection between p-adic characters and sequences of elements in Zq, indexed by natural numbers relatively prime to p, and for which the limit is zero. To such a p-adic character we associate an L-function, and we prove that this L-function is p-adic meromorphic if the corresponding sequence is overconvergent. If more generally the sequence is Clog-convergent, we show that the associated L-function is meromorphic in the open disk of radius qC . Finally, we exhibit examples of Clog-convergent sequences with associated L-functions which are not meromorphic in the disk of radius qC+ε for any ε > 0.