Are there any statements in ancient Jewish literature that refer to the dead as being like angels?
Angels are beings that are created for a specific purpose. People are created with free will and establish their own level. This concept is discussed in a number of questions such as What is the purpose of life on this earth and Is free will limited to humans? and Jewish concept of angels and Is free will limited to humans? and Sources for free will of angels
I put the word and between each link so that you can see that they are separate.
However, this only applies to while people are alive. After death, they can no longer perform mitzvos and they have established the level at which they will exist for eternity. This is one of the concepts of "reward and punishment". Thus, after a person dies, they no longer have the ability to change (like angels), but they are not as "robotic" as angels which were created for a specific task and only exist as long as that task has not yet been completed.