In this study, we show that every continuous Jordan left derivation on a (commutative or noncommutative) prime UMV-Banach algebra with the identity element 1 is identically zero. Moreover, we prove that every continuous left derivation on a unital finite dimensional Banach algebra, under certain conditions, is identically zero. As another result in this regard, it is proved that if R is a 2-torsion free semiprime ring such that annf[y; z] j y; z 2 Rg = f0g, then every Jordan left derivation L : R ! R is identically zero. In addition, we provide several other results in this regard.