This paper proposes a method to effectively perform joint training-and-pruning based on adaptive dropout layers with unit-wise retention probabilities. The proposed method is based on the estimation of a unit-wise retention probability in a dropout layer. A unit that is estimated to have a small retention probability can be considered to be prunable. The retention probability of the unit is estimated using back-propagation and the Gumbel-Softmax technique. This pruning method is applied at several application points in Conformers such that the effective number of parameters can be significantly reduced. Specifically, adaptive dropout layers are introduced in three locations in each Conformer block: (a) the hidden layer of the feed-forward-net component, (b) the query vectors and the value vectors of the self-attention component, and (c) the input vectors of the LConv component. The proposed method is evaluated by conducting a speech recognition experiment on the LibriSpeech task. It was shown that this approach could simultaneously achieve a parameter reduction and accuracy improvement. The word error rates improved by approx 1% while reducing the number of parameters by 54%.