The chaser and the chased can offer Salat (prayer) while riding , and by signs
بَابُ صَلاَةِ الطَّالِبِ وَالْمَطْلُوبِ رَاكِبًا وَإِيمَاءً
Sahih al-Bukhari, Book of Fear Prayer(view book)
Sahih al-Bukhari 946
Narrated Jabir bin 'Abdullah:
On the day of the Khandaq Umar came, cursing the disbelievers of Quraish and said, "O Allah's Apostle! I have not offered the 'Asr prayer and the sun has set." The Prophet replied, "By Allah! I too, have not offered the prayer yet. "The Prophet then went to Buthan, performed ablution and performed the 'Asr prayer after the sun had set and then offered the Maghrib prayer after it."
حَدَّثَنَا يَحْيَى، قَالَ حَدَّثَنَا وَكِيعٌ، عَنْ عَلِيِّ بْنِ مُبَارَكٍ، عَنْ يَحْيَى بْنِ أَبِي كَثِيرٍ، عَنْ أَبِي سَلَمَةَ، عَنْ جَابِرِ بْنِ عَبْدِ اللَّهِ، قَالَ جَاءَ عُمَرُ يَوْمَ الْخَنْدَقِ، فَجَعَلَ يَسُبُّ كُفَّارَ قُرَيْشٍ وَيَقُولُ يَا رَسُولَ اللَّهِ مَا صَلَّيْتُ الْعَصْرَ حَتَّى كَادَتِ الشَّمْسُ أَنْ تَغِيبَ. فَقَالَ النَّبِيُّ ﷺ " وَأَنَا وَاللَّهِ مَا صَلَّيْتُهَا بَعْدُ ". قَالَ فَنَزَلَ إِلَى بُطْحَانَ فَتَوَضَّأَ، وَصَلَّى الْعَصْرَ بَعْدَ مَا غَابَتِ الشَّمْسُ، ثُمَّ صَلَّى الْمَغْرِبَ بَعْدَهَا.