Can the dendrites of sensory neurons be a meter long?
Short answer
Axons can be over a meter long, but dendrites are never that long. Distance in the body is covered by axons. The dendritic part of skin receptors is generally considered to be the receptor part and the receptor part only. The elongated structure leading to the soma, as well as the axonal part to the spinal cord are generally considered to be one and the same axon, the soma being attached to the axon in this case.

Background
The sensory receptors in the skin have their cell bodies located in the dorsal root ganglia situated adjacent of the spinal cord (Bourinet et al., 2014). In the case of pain receptors it is shown in Fig. 1. The same basic structure holds for the touch receptors in the skin.


Fig. 1. Signaling of pain receptors in the skin. source: Bourinet et al. (2014)  

The cell bodies are located in the dorsal root ganglia of the spinal cord. Action potentials are sent from the dendritic region in the skin to the cell body in the ganglion. From there on it is transmitted to spinal interneurons in the spinal cord and up to the brain.

The nomenclature of these sensory neurons is generally as follows: in the cell body the axon splits in two. The afferent part that transmits the pulses from the skin to the cell body and the part that transmits the signal from the soma away from the cell body to the spinal cord are both considered as a part of the axon. The dendritic region is generally considered to be the nerve ending (Fig. 2C). So the axons of peripheral sensory neurons, as well as motor neurons for that matter, can indeed be over a meter long (Lodish et al., 2000).


Fig. 2. Types of neurons. Arrows indicate the direction of conduction of action potentials in axons. (A) Multipolar interneurons. The dendrites receive signals at synapses with several hundred other neurons, and there is a single long branching axon. (B) A motor neuron innervating a muscle cell. Typically, motor neurons have a single long axon extending from the cell body to the effector cell. (C) A sensory neuron in which the axon branches just after it leaves the cell body. The peripheral branch carries the nerve impulse from the receptor cell to the cell body, which is located in the dorsal root ganglion near the spinal cord; the central branch carries the impulse from the cell body to the spinal cord or brain. Both branches are structurally and functionally axons, except at their terminal portions, even though the peripheral branch conducts impulses toward, rather than away from, the cell body. source: Lodish et al. (2000)   

References
- Bourinet et al., Physiol Rev (2014); 94(1): 81-140
- Lodish et al., Molecular Cell Biology, 4th ed. New York: Freeman (2000)