To answer this question, we need to know
a) What exactly is a 'living being'?
b) Did "live" exist before single celled organisms.
In Islam, there are instances of non-sentient beings being also classified as living creatures. For clarification, as per science, an individual form of life, composed of a single cell or a complex of cells in which organelles work together to carry out the various processes of life is defined to be living. If it also has a central nervous system and the capacity to perceive an environment and think through reactions, it is thought to be a sentient being.
But there are various ayahs in the Quran and many Sahih Hadiths where things like rocks, the sun, the moon, Heaven, Hell, the Pen and even our planet Earth are represented as living, talking creatures.
Infact, there is a collection of Hadiths that state that even death will die at the end of time implying it is in some way form of life.
Hence, it is difficult to Islamically classify what exactly is a 'living being'.
Coming to the second part of your question, from the Hadiths, we understand that Angels existed before the creation of our universe (under the seventh sky). So life did exist before the emergence of single celled organisms on our planet.
Also with reference to your comment of being created from water, in the Quran, it is stated that humans are created from a 'nutfah' (a small drop of fluid)
While there is no direct reference to the concept of evolution in Islam, many modern day ulema tend to state the even if evolution is true, it doesn't invalidate any concept of Islam as Allah is capable to both have evolution among all earthly creatures AND also create the Angels, Jinns and Humans directly without evolution.
The story of the creation of Adam is well explained in the Quran. Also the create of Eve and the birth of Prophet Jesus are testament to the Power of Allah
Also look into the this answer by @hkBattousai