Strain is defined as change in dimension over original dimension. Now we know dimension has the units of L thus this means it has no dimensional formula and has no units.
Strain= Change in dimension/Original dimension
Thus
Dimensional Formula of Strain= (M0L1T0)/(M0L1T0)
Dimensional Formula of Strain= M0L0T0
Now we know dimension has the units of L thus same units cancel each other leaving numeric value only. So Strain has no SI unit.