Yes, there are P-complete problems just as there are NP-complete problems. Wikipedia has an article on this, which gives several examples, such as the circuit value problem (determining whether a given Boolean formula is true or false for given values of the variables).
However, the existence of such problems doesn't necessarily give you new, more efficient algorithms for problems in P. A simple version of the time hierarchy theorem says that there are problems that, for all constants~$k$, can be solved in time $O(n^{k+1})$ that cannot be solved in time $O(n^k)$. Using reductions doesn't get around this. For example, to solve a problem that requires time $n^5$ by reduction to a complete problem that can be solved in time $n^2$ would require a reduction that increased the length of the input to about $n^{5/2}\!$, and the whole process would therefore still take time $n^5\!$.
As usual, there's no free lunch.