Are any problems in P known to have logarithmic-time-verifiable certificates?
The question here is that in the certificate definition of $\mathsf{NP}$, what is more important is the size of the certificate (in the size of the input), the complexity class that  the certificate verifier belongs too is less important. In fact you can take the verifier to be in $\mathsf{AC^0}$ as long as the size of the certificates are polynomial in the size of the input and you will still get the whole problems in $\mathsf{NP}$. $\mathsf{AC^0}$ is a very small complexity class that is included in almost every complexity class you have heard of.

To obtain smaller classes you also need to restrict the way the certificate verifier accesses the certificate. One kind of restriction that is considered is restricting the number of times  the verifier can read the input using a log-space verifier. Something similar may work to capture $\mathsf{P}$, but just capturing it in this way is not interesting. The certificate definition should be nice and interesting so that it would give us some insight about $\mathsf{P}$. The certificate definition of $\mathsf{NP}$ is interesting because it is an exceptionally nice, it is even more intuitive than the original definition using non-intuitive non-deterministic machines. It allows us to cast the question regarding $\mathsf{P}$ vs. $\mathsf{NP}$ as "efficient solving vs. efficient verifying". Just having a certificate definition by itself is not very interesting.