A

Step 1: Understand the problem. We have a chain of length λ and mass m being pulled up over the edge of a table. Only a portion of the chain will hang off the edge during the process.

Step 2: Calculate the weight of the hanging portion. As the chain is pulled up, if a length x of the chain is hanging off the edge, the mass of the hanging portion is given by the equation:
$$ m_{hanging} = m \cdot \frac{x}{\lambda} $$

Step 3: Calculate the force acting on the hanging portion due to gravity. The gravitational force on the hanging portion is:
$$ F_{gravity} = m_{hanging} \cdot g = \left(m \cdot \frac{x}{\lambda}\right) g $$

Step 4: As the chain is pulled at a constant speed, the pulling force (F) should balance the weight of the hanging chain.

Step 5: Work is defined as the force times the distance. In this case, since we're raising the chain, the total work done, W, to pull the whole chain length λ upwards is:
$$ W = F \cdot d $$
where F is the force needed to lift the entire weight of the chain and d is the distance moved, which equals the total length of the chain, λ.

Step 6: Calculate the total weight of the chain. The force needed to lift the entire chain is:
$$ F = m \cdot g $$

Step 7: Therefore, the work done in pulling the entire length of the chain up to the table is:
$$ W = (m \cdot g) \cdot \lambda $$

Thus, the work done by the force till the chain reaches the horizontal surface of the table is \( m g \lambda \).

Therefore, the answer is option A.