Did Jesus have an immortal soul?

Firstly, “soul” just means “living being”, so people do not have souls, they are living souls. See How is “soul” used in the Bible?.

Secondly, Jesus died, so he clearly was not immortal, or he did not die.  See Was Jesus really dead for three days? “I was dead” (Revelation 1:18)

The Lord Jesus was only made immortal by his Father, God, after God raised Jesus from the dead (1 Cor 15:42-49). So Jesus has immortality now.