We need to agree on a definition of the soul, or at least an observable aspect of it. I would say that one such aspect is having feelings.
Judging by this aspect, the answer is yes. Seerah tells us they do. You may read about the story of the tree that felt sad when the messenger (PBUH) left it. The story and hadith are in Arabic here: http://library.islamweb.net/newlibrary/display_book.php?idfrom=71&idto=71&bk_no=118&ID=77
Relevant ahadith:
- In Sahih al-Bukhari:
Narrated Anas bin Malik:
That he heard Jabir bin `Abdullah saying, "The roof of the Mosque was built over trunks of datepalms working as pillars. When the Prophet (ﷺ) delivered a sermon, he used to stand by one of those trunks till the pulpit was made for him, and he used it instead. Then we heard the trunk sending a sound like of a pregnant she-camel till the Prophet (ﷺ) came to it, and put his hand over it, then it became quiet."
حَدَّثَنَا إِسْمَاعِيلُ، قَالَ حَدَّثَنِي أَخِي، عَنْ سُلَيْمَانَ بْنِ بِلاَلٍ، عَنْ يَحْيَى بْنِ سَعِيدٍ، قَالَ أَخْبَرَنِي حَفْصُ بْنُ عُبَيْدِ اللَّهِ بْنِ أَنَسِ بْنِ مَالِكٍ، أَنَّهُ سَمِعَ جَابِرَ بْنَ عَبْدِ اللَّهِ ـ رضى الله عنهما ـ يَقُولُ كَانَ الْمَسْجِدُ مَسْقُوفًا عَلَى جُذُوعٍ مِنْ نَخْلٍ فَكَانَ النَّبِيُّ صلى الله عليه وسلم إِذَا خَطَبَ يَقُومُ إِلَى جِذْعٍ مِنْهَا، فَلَمَّا صُنِعَ لَهُ الْمِنْبَرُ، وَكَانَ عَلَيْهِ فَسَمِعْنَا لِذَلِكَ الْجِذْعِ صَوْتًا كَصَوْتِ الْعِشَارِ، حَتَّى جَاءَ النَّبِيُّ صلى الله عليه وسلم فَوَضَعَ يَدَهُ عَلَيْهَا فَسَكَنَتْ. - In Jami` at-Tirmidhi