We report an analog neuromorphic module composed of p-type carbon nanotube (CNT) synapses and an integrate-and-fire (I&F) circuit. The CNT synapse has a field-effect transistor structure with a random CNT network as its channel and an aluminum oxide dielectric layer implanted with indium ions as its gate. A positive voltage pulse (spike) applied on the gate attracts electrons into the defect sites of the gate dielectric layer, and the trapped electrons are gradually released after the pulse is removed. The electrons modify the hole concentration and induce a dynamic postsynaptic current in the CNT channel. Multiple input spikes induce excitatory or inhibitory postsynaptic currents via excitatory or inhibitory CNT synapses, which flow toward an I&F circuit to trigger output spikes. The dynamic transfer function between the input and output spikes of the neuromorphic module is analyzed. The module could potentially be scaled up to emulate biological neural networks and their functions.